sec^2(3θ)cos^2(3θ)=sec^2(3θ)+cos^2(3θ)

{ "query": { "display": "$$\\sec^{2}\\left(3θ\\right)\\cos^{2}\\left(3θ\\right)=\\sec^{2}\\left(3θ\\right)+\\cos^{2}\\left(3θ\\right)$$", "symbolab_question": "EQUATION#\\sec^{2}(3θ)\\cos^{2}(3θ)=\\sec^{2}(3θ)+\\cos^{2}(3θ)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:θ\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sec^{2}\\left(3θ\\right)\\cos^{2}\\left(3θ\\right)=\\sec^{2}\\left(3θ\\right)+\\cos^{2}\\left(3θ\\right){\\quad:\\quad}$$No Solution for $$θ\\in\\mathbb{R}$$", "input": "\\sec^{2}\\left(3θ\\right)\\cos^{2}\\left(3θ\\right)=\\sec^{2}\\left(3θ\\right)+\\cos^{2}\\left(3θ\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\sec^{2}\\left(3θ\\right)+\\cos^{2}\\left(3θ\\right)$$ from both sides", "result": "\\sec^{2}\\left(3θ\\right)\\cos^{2}\\left(3θ\\right)-\\sec^{2}\\left(3θ\\right)-\\cos^{2}\\left(3θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-\\cos^{2}\\left(3θ\\right)-\\sec^{2}\\left(3θ\\right)+\\cos^{2}\\left(3θ\\right)\\sec^{2}\\left(3θ\\right)", "result": "1-\\frac{1}{\\sec^{2}\\left(3θ\\right)}-\\sec^{2}\\left(3θ\\right)=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cos\\left(x\\right)=\\frac{1}{\\sec\\left(x\\right)}$$", "result": "=-\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}-\\sec^{2}\\left(3θ\\right)+\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}\\sec^{2}\\left(3θ\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}-\\sec^{2}\\left(3θ\\right)+\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}\\sec^{2}\\left(3θ\\right):{\\quad}-\\frac{1}{\\sec^{2}\\left(3θ\\right)}-\\sec^{2}\\left(3θ\\right)+1$$", "input": "-\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}-\\sec^{2}\\left(3θ\\right)+\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}\\sec^{2}\\left(3θ\\right)", "result": "=-\\frac{1}{\\sec^{2}\\left(3θ\\right)}-\\sec^{2}\\left(3θ\\right)+1", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}=\\frac{1}{\\sec^{2}\\left(3θ\\right)}$$", "input": "\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{\\sec^{2}\\left(3θ\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{\\sec^{2}\\left(3θ\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qtWSHifc9AkvQ1rSvIucJAS9Lsqn5/hbZ79ZYpsbMkzehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpnCqjrVBfSeawkE9dsrSsYUAqZi/IhnrNgdk/vKnGsjG/HXd6OvxDi4aYZMSLkMcpVmedLvEA4iKL64E6l13BMNU3N7RkMqhonJ6vldzL5W9uv0ixGgyIRhbSQwpmgjshw==" } }, { "type": "interim", "title": "$$\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}\\sec^{2}\\left(3θ\\right)=1$$", "input": "\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}\\sec^{2}\\left(3θ\\right)", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}=\\frac{1}{\\sec^{2}\\left(3θ\\right)}$$", "input": "\\left(\\frac{1}{\\sec\\left(3θ\\right)}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{\\sec^{2}\\left(3θ\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{\\sec^{2}\\left(3θ\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qtWSHifc9AkvQ1rSvIucJAS9Lsqn5/hbZ79ZYpsbMkzehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpnCqjrVBfSeawkE9dsrSsYUAqZi/IhnrNgdk/vKnGsjG/HXd6OvxDi4aYZMSLkMcpVmedLvEA4iKL64E6l13BMNU3N7RkMqhonJ6vldzL5W9uv0ixGgyIRhbSQwpmgjshw==" } }, { "type": "step", "result": "=\\frac{1}{\\sec^{2}\\left(3θ\\right)}\\sec^{2}\\left(3θ\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\sec^{2}\\left(3θ\\right)}{\\sec^{2}\\left(3θ\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sec^{2}\\left(3θ\\right)$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qtWSHifc9AkvQ1rSvIucJLG6E67+oX7rOjaaQnAxs3OJESRKAPnPTpzW/5HCFtnK3XeO2tIUPH5Q2xrCOU6NXdm4KC5o252PmptjUrwD59LfRaH3KOLhpljC85zkkN7NB0fQZBfFM+hd6e4+R58dL+IKM4zQrhDxuQn3sB7yr/3rb2OI7vxJ1F08QFWEXIMS" } }, { "type": "step", "result": "=-\\frac{1}{\\sec^{2}\\left(3θ\\right)}-\\sec^{2}\\left(3θ\\right)+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7V38ra/PyA99tfFw+73cYuYET1oaVgMTcOSWU7q8tyZ2i5W0pj2l8V3szrsyqTAl4NFMVpvgMBjD05g6CNuOV9nPtmVOqxcG75DTCOt8x+axePM4Tl9XrDsir/L1SwoVUCUCWbkwGOY7PqKo3U/JLJRj26C7VDCYE+7Mj803s/VL1rFCnlx61fwiPKPZl/wNd7AM0AwbbEzaGPZaw+bJsLmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM96jKrzzhOmB/zFA60m7UioIET1oaVgMTcOSWU7q8tyZ2i5W0pj2l8V3szrsyqTAl4NFMVpvgMBjD05g6CNuOV9nPtmVOqxcG75DTCOt8x+aylxX/i9o5y+TEeEP36Y0NB" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sxSOfOl7Um+zqSbarNITtmMjQuCJwzQScIYjlLxErelgewk3mcYxiIo0PZy7cfvz9gY4gOJ+t+YTbNsqOcCXRlwmA5/FoiLOOsN2I6qT0qWAYUxCRRMw2cScUTF07kq8MvMnPiuRieO/X9beCVqRKbWzVFA6VxLY8LrAw/d4eKsVAy9QTupoWSUuf53i1qjLkRclFFKqHLnxcdekeCC98gcl0JuOqOgr6zQAyRhtIaBkS3dlcCKpQTQcheuut7MkmLRx2SCfDYc+NFWlwczFlshyJhXONoCiURo2RU420EFGmlM2AwWiez1syRPMgeig" } }, { "type": "interim", "title": "Solve by substitution", "input": "1-\\frac{1}{\\sec^{2}\\left(3θ\\right)}-\\sec^{2}\\left(3θ\\right)=0", "result": "\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i", "steps": [ { "type": "step", "primary": "Let: $$\\sec\\left(3θ\\right)=u$$", "result": "1-\\frac{1}{u^{2}}-u^{2}=0" }, { "type": "interim", "title": "$$1-\\frac{1}{u^{2}}-u^{2}=0{\\quad:\\quad}u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i$$", "input": "1-\\frac{1}{u^{2}}-u^{2}=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$u^{2}$$", "input": "1-\\frac{1}{u^{2}}-u^{2}=0", "result": "u^{2}-1-u^{4}=0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$u^{2}$$", "result": "1\\cdot\\:u^{2}-\\frac{1}{u^{2}}u^{2}-u^{2}u^{2}=0\\cdot\\:u^{2}" }, { "type": "interim", "title": "Simplify", "input": "1\\cdot\\:u^{2}-\\frac{1}{u^{2}}u^{2}-u^{2}u^{2}=0\\cdot\\:u^{2}", "result": "u^{2}-1-u^{4}=0", "steps": [ { "type": "interim", "title": "Simplify $$1\\cdot\\:u^{2}:{\\quad}u^{2}$$", "input": "1\\cdot\\:u^{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:u^{2}=u^{2}$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bSuyvEnfFkHF7+RrUJqlmS061ljBSPJeENOw2efoSWtza5guT/QQx17cJgH2E/w7/z//r+dXk7h9vxeDCLuZqnKF3u2OIb4bFA3EO8aRlSUrHg5YhigxLFTF32Kxs0sUialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$-\\frac{1}{u^{2}}u^{2}:{\\quad}-1$$", "input": "-\\frac{1}{u^{2}}u^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:u^{2}}{u^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$u^{2}$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I5jNqiJ9xF0G3MZkdIfbCN8q7iYRc0ARJoPbvuRgXs11g99dC9fj9sg0EHzBIRDRlcq1iPPbKQKUi0Yqft4tTk3kCh3oevUunZ7/b0qFKBQ4EZtqXS+QoXv8EnlO8In44u6xgtrkRURyIPTnk8RHRLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Simplify $$-u^{2}u^{2}:{\\quad}-u^{4}$$", "input": "-u^{2}u^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$u^{2}u^{2}=\\:u^{2+2}$$" ], "result": "=-u^{2+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=-u^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sDAOWLoiusxs+0TJOF3d096GQqufR6tr2vPxOUv7H+/98MZsA86MuvbmLMKeJay+P8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXT9uQZA5MoPH4yl9dXQZPrFJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Simplify $$0\\cdot\\:u^{2}:{\\quad}0$$", "input": "0\\cdot\\:u^{2}", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vATyfz+2OhYcpP2syCG80y061ljBSPJeENOw2efoSWtRZPRrfkNDmi+szkABFipURSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pS+x3wex/rZPGJ7Ox98r4U" } }, { "type": "step", "result": "u^{2}-1-u^{4}=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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$$u^{2}-1-u^{4}=0:{\\quad}u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i$$", "input": "u^{2}-1-u^{4}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$a_{n}x^{n}+\\ldots\\:+a_{1}x+a_{0}=0$$", "result": "-u^{4}+u^{2}-1=0" }, { "type": "step", "primary": "Rewrite the equation with $$x=u^{2}$$ and $$x^{2}=u^{4}$$", "result": "-x^{2}+x-1=0" }, { "type": "interim", "title": "Solve $$-x^{2}+x-1=0:{\\quad}x=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:x=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}$$", "input": "-x^{2}+x-1=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-x^{2}+x-1=0", "result": "{x}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\left(-1\\right)}}{2\\left(-1\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-1,\\:b=1,\\:c=-1$$", "result": "{x}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\left(-1\\right)}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": 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$$\\sqrt{1^{2}-4\\left(-1\\right)\\left(-1\\right)}:{\\quad}\\sqrt{3}i$$", "input": "\\sqrt{1^{2}-4\\left(-1\\right)\\left(-1\\right)}", "result": "{x}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{3}i}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\left(-1\\right)\\left(-1\\right)}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1-4\\cdot\\:1\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=\\sqrt{1-4}" }, { "type": "step", "primary": "Subtract the numbers: $$1-4=-3$$", "result": "=\\sqrt{-3}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$", "secondary": [ "$$\\sqrt{-3}=\\sqrt{-1}\\sqrt{3}$$" ], "result": "=\\sqrt{-1}\\sqrt{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=\\sqrt{3}i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-1+\\sqrt{3}i}{2\\left(-1\\right)},\\:{x}_{2}=\\frac{-1-\\sqrt{3}i}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$x=\\frac{-1+\\sqrt{3}i}{2\\left(-1\\right)}:{\\quad}\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{-1+\\sqrt{3}i}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1+\\sqrt{3}i}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1+\\sqrt{3}i}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{-1+\\sqrt{3}i}{2}" }, { "type": "interim", "title": "Rewrite $$-\\frac{-1+\\sqrt{3}i}{2}$$ in standard complex form: $$\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i$$", "input": "-\\frac{-1+\\sqrt{3}i}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{-1+\\sqrt{3}i}{2}=-\\left(-\\frac{1}{2}\\right)-\\left(\\frac{\\sqrt{3}i}{2}\\right)$$" ], "result": "=-\\left(-\\frac{1}{2}\\right)-\\left(\\frac{\\sqrt{3}i}{2}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}-\\frac{\\sqrt{3}i}{2}" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS9z4F6b9EBEhpTHCu4QIJPQnVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZwPTM+xnhoGkKkgIQ0VhzmNz4F6b9EBEhpTHCu4QIJPRMYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0=" } }, { "type": "step", "result": "=\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gI32wa8JgqOZw2m0j1OS942If1DMR0BXBjsl6yqn0GZV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAll3Hst1hXeka5lMz6hutiM6HbENYkHXOlwEG2WAaSblmLdiDG/FnL4VjCkVGdCOoquPwJ+pefpTeaFGZC/bAr3rmy8TLV3HzpkwVsf8eoxsn" } }, { "type": "interim", "title": "$$x=\\frac{-1-\\sqrt{3}i}{2\\left(-1\\right)}:{\\quad}\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{-1-\\sqrt{3}i}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1-\\sqrt{3}i}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1-\\sqrt{3}i}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{-1-\\sqrt{3}i}{2}" }, { "type": "interim", "title": "Rewrite $$-\\frac{-1-\\sqrt{3}i}{2}$$ in standard complex form: $$\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i$$", "input": "-\\frac{-1-\\sqrt{3}i}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{-1-\\sqrt{3}i}{2}=-\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{3}i}{2}\\right)$$" ], "result": "=-\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{3}i}{2}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}+\\frac{\\sqrt{3}i}{2}" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS1BGfus/0qXQoc6ObyzUa30nVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZwPTM+xnhoGkKkgIQ0VhzmFBGfus/0qXQoc6ObyzUa31MYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0=" } }, { "type": "step", "result": "=\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LcON8lIS+hibufdR4EzXnI2If1DMR0BXBjsl6yqn0GZV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAllLGaFm81fCELCmS+iMenqSHbENYkHXOlwEG2WAaSblmLdiDG/FnL4VjCkVGdCOoqlOzjFFhwU2jBC8En0RaGbDmy8TLV3HzpkwVsf8eoxsn" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:x=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:x=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Substitute back $$x=u^{2},\\:$$solve for $$u$$" }, { "type": "interim", "title": "Solve $$u^{2}=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}:{\\quad}u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i$$", "input": "u^{2}=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Substitute $$u=x+yi$$", "result": "\\left(x+yi\\right)^{2}=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Expand $$\\left(x+yi\\right)^{2}:{\\quad}\\left(x^{2}-y^{2}\\right)+2ixy$$", "input": "\\left(x+yi\\right)^{2}", "result": "\\left(x^{2}-y^{2}\\right)+2ixy=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "result": "=\\left(x+iy\\right)^{2}" }, { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=yi$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}+2xyi+\\left(yi\\right)^{2}" }, { "type": "interim", "title": "$$\\left(yi\\right)^{2}=-y^{2}$$", "input": "\\left(yi\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=i^{2}y^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$i^{2}=-1$$", "input": "i^{2}", "steps": [ { "type": "step", "primary": "Apply imaginary number rule: $$i^{2}=-1$$", "result": "=-1", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ixd6+oXBeBbBYSxiL8OAvAlAlm5MBjmOz6iqN1PySyUyzPuBwg0tEkeKHbqH8v7p1V4jc9gN7k5UVTYGDf0b7CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(-1\\right)y^{2}" }, { "type": "step", "primary": "Refine", "result": "=-y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76BIpiHUJqX+Gjz6Fwr0+9c0ag8T1MwTer44+aCS/ZFC+KWdrKsfvNYjtpYgUTWx00q1f0AvAEP8N09ISyDIZLcjMJz/TOsacQU4ZUGiOLLI=" } }, { "type": "step", "result": "=x^{2}+2ixy-y^{2}" }, { "type": "interim", "title": "Rewrite $$x^{2}+2ixy-y^{2}$$ in standard complex form: $$\\left(x^{2}-y^{2}\\right)+2xyi$$", "input": "x^{2}+2ixy-y^{2}", "steps": [ { "type": "step", "primary": "Group the real part and the imaginary part of the complex number", "result": "=\\left(x^{2}-y^{2}\\right)+2xyi" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fyAweyQvVwJsSQQKDjm97PG9YG39LAdblepQizswpHwa2CjokqvmBtYUm1DzxmMxKwNwzwWuFelyz8uvoD3mF2RLd2VwIqlBNByF6663sySYtHHZIJ8Nhz40VaXBzMWWxf2ToaH1kMvtQ3yodc/G8DRwLaryLWYBUQDrY2Dn8OshJnnwHxnMGhyjcgBVVR+ihbXRsjO6u6m4ZMlRD4wxtQ==" } }, { "type": "step", "result": "=\\left(x^{2}-y^{2}\\right)+2xyi" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7acBL/XWGdfMl4uhsCDRDUqr/LfvPjgKL/AHdgs1FR4p1DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhlGg7dnDDOhlC500PDgRQdo+i0Ux3lprvX50CFfl5rrAQ==" } }, { "type": "step", "primary": "Complex numbers can be equal only if their real and imaginary parts are equal", "secondary": [ "Rewrite as system of equations:" ], "result": "\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=-\\frac{\\sqrt{3}}{2}\\end{bmatrix}" }, { "type": "interim", "title": "$$\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=-\\frac{\\sqrt{3}}{2}\\end{bmatrix}:{\\quad}\\begin{pmatrix}x=-\\frac{\\sqrt{3}}{2},\\:&y=\\frac{1}{2}\\\\x=\\frac{\\sqrt{3}}{2},\\:&y=-\\frac{1}{2}\\end{pmatrix}$$", "input": "\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=-\\frac{\\sqrt{3}}{2}\\end{bmatrix}", "steps": [ { "type": "interim", "title": "Isolate $$x\\:$$for $$2xy=-\\frac{\\sqrt{3}}{2}:{\\quad}x=-\\frac{\\sqrt{3}}{4y}$$", "input": "2xy=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2y$$", "input": "2xy=-\\frac{\\sqrt{3}}{2}", "result": "x=-\\frac{\\sqrt{3}}{4y}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2y$$", "result": "\\frac{2xy}{2y}=\\frac{-\\frac{\\sqrt{3}}{2}}{2y}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2xy}{2y}=\\frac{-\\frac{\\sqrt{3}}{2}}{2y}", "result": "x=-\\frac{\\sqrt{3}}{4y}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2xy}{2y}:{\\quad}x$$", "input": "\\frac{2xy}{2y}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\frac{xy}{y}" }, { "type": "step", "primary": "Cancel the common factor: $$y$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aLZIDEtiNWvyIQ1h+ZnJcnyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz3vPC+dAdNgr+i2Xzs9Nk6DialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{\\sqrt{3}}{2}}{2y}:{\\quad}-\\frac{\\sqrt{3}}{4y}$$", "input": "\\frac{-\\frac{\\sqrt{3}}{2}}{2y}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{\\sqrt{3}}{2}}{2y}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{\\sqrt{3}}{2}}{2y}=\\frac{\\sqrt{3}}{2\\cdot\\:2y}$$" ], "result": "=-\\frac{\\sqrt{3}}{2\\cdot\\:2y}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=-\\frac{\\sqrt{3}}{4y}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVTVQn0Dd+FxZDRYQPY4AlXTOh8i9AGblNorp5llrtuk6o5FYteSPKwXny4uCMrdsKwXnBLrNHLPx6zrUIO/1ivRtOgK3JSit4ul3FxeB4TlDHjb2+5NLFZrsH9fcPWg/TcBZCZf4UBlDnt5DwfBsIkIF5wS6zRyz8es61CDv9Yr0pkI3O5C/AOSITxbWmWxWwQ==" } }, { "type": "step", "result": "x=-\\frac{\\sqrt{3}}{4y}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "Plug the solutions $$x=-\\frac{\\sqrt{3}}{4y}$$ into $$x^{2}-y^{2}=\\frac{1}{2}$$" }, { "type": "interim", "title": "For $$x^{2}-y^{2}=\\frac{1}{2}$$, subsitute $$x$$ with $$-\\frac{\\sqrt{3}}{4y}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "steps": [ { "type": "step", "primary": "For $$x^{2}-y^{2}=\\frac{1}{2}$$, subsitute $$x$$ with $$-\\frac{\\sqrt{3}}{4y}$$", "result": "\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}" }, { "type": "interim", "title": "Solve $$\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Multiply by LCM", "input": "\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}", "result": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}:{\\quad}\\frac{3}{16y^{2}}$$", "input": "\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}", "result": "\\frac{3}{16y^{2}}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}=\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}$$" ], "result": "=\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{\\left(4y\\right)^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(4y\\right)^{2}=4^{2}y^{2}$$" ], "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{4^{2}y^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3}{4^{2}y^{2}}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\frac{3}{16y^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75s3xpOjgqYGSPN+T51/sNd8Wehz8oJ6Q0nZ9rYg/w1tV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlj8+Ql4PFSdVJ69xUTzBMi1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qhmpS/ZcIy9T4S6pZAcIqo11ScG+o50Rl37IDTs85LWy" } }, { "type": "interim", "title": "Find Least Common Multiplier of $$16y^{2},\\:2:{\\quad}16y^{2}$$", "input": "16y^{2},\\:2", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Least Common Multiplier of $$16,\\:2:{\\quad}16$$", "input": "16,\\:2", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$16:{\\quad}2\\cdot\\:2\\cdot\\:2\\cdot\\:2$$", "input": "16", "steps": [ { "type": "step", "primary": "$$16\\:$$divides by $$2\\quad\\:16=8\\cdot\\:2$$", "result": "=2\\cdot\\:8" }, { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdy5Ljmr36t3AP8UKimRmJwFB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECWYh2nZH9X0Rq5Oe4GdfGo" } }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$16$$ or $$2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:2\\cdot\\:2=16$$", "result": "=16" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$16y^{2}$$ or $$2$$", "result": "=16y^{2}" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top in Equation Title 1Eq" } }, { "type": "step", "primary": "Multiply by LCM=$$16y^{2}$$", "result": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}-y^{2}\\cdot\\:16y^{2}=\\frac{1}{2}\\cdot\\:16y^{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}-y^{2}\\cdot\\:16y^{2}=\\frac{1}{2}\\cdot\\:16y^{2}", "result": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3}{16y^{2}}\\cdot\\:16y^{2}:{\\quad}3$$", "input": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:16y^{2}}{16y^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$16$$", "result": "=\\frac{3y^{2}}{y^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$y^{2}$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iN4dvGcm+bQGP3fStnffSNsO8nDHBnMsuZu4AFRSIkBZ8mEc9fB7wfq6et5j4rXPgeXEFCD58rDFDfybZf16JD/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10YboD4oQjQoml9oHnDEg4Hb6n+FJ66//gnBohluY6VGj6LRTHeWmu9fnQIV+XmusB" } }, { "type": "interim", "title": "Simplify $$-y^{2}\\cdot\\:16y^{2}:{\\quad}-16y^{4}$$", "input": "-y^{2}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$y^{2}y^{2}=\\:y^{2+2}$$" ], "result": "=-16y^{2+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=-16y^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s708LaLmsXDBKqBgcrTGi8+m+2jZ3RVS4dK1SlxcsDTtWrju+5Z51e/ZZSD3gRHwjBCLyOc3zohAErAg63DYY3+4EFMST8lDZxn1Yq5HMKVTs6Zt5XsJOFJ/EboUEQ+/cI/hKRA7B3zygP4XVuJ6GHk48BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:16y^{2}:{\\quad}8y^{2}$$", "input": "\\frac{1}{2}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:16}{2}y^{2}" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:16}{2}=8$$", "input": "\\frac{1\\cdot\\:16}{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:16=16$$", "result": "=\\frac{16}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{16}{2}=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCmUJCqRgq/EpL80lkhI7O+YJQJZuTAY5js+oqjdT8ksl2+Bm4xvhfGGRR5Jko7aCV/QVTV4d0XUrkjGhi3/crx0OteLnHolAnrbIwv7NbB+T" } }, { "type": "step", "result": "=8y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8crqNKUhefhKe0Zr7BxGuZQgJ/ZZA32ZInFBpDtxBfiKQdmmBkyA4nheYWVD5Bw8o2RLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9d4FJuPgO/OHNuxTBtnbIQjyxDSHcil1+wqic2arrKOM=" } }, { "type": "step", "result": "3-16y^{4}=8y^{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Equation LCM Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaQjhXqywo+3heGYONXKDTxsYRCQ1RVKa+ODKHx99mkMHsAdjBhHw+s7du5jhEYrzVLyn2EcMq920q9Gc88uT7GKw6xR4edfo6toG7p6T0x0nOGnlfuQCWLBLhcJF31TRtbcrHqEfjIkjv9BdwwRsdHWgCnLxyWjXu0432/oyzYcd/YctE2oLW2snqdZCVuig=" } }, { "type": "interim", "title": "Solve $$3-16y^{4}=8y^{2}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Move $$8y^{2}\\:$$to the left side", "input": "3-16y^{4}=8y^{2}", "result": "3-16y^{4}-8y^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$8y^{2}$$ from both sides", "result": "3-16y^{4}-8y^{2}=8y^{2}-8y^{2}" }, { "type": "step", "primary": "Simplify", "result": "3-16y^{4}-8y^{2}=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fn8/0AnhCyqGUcjKQ6pS92xJ8OaIWFMGjlQIFe1Bhhjyxv83nWcSQlsRHYjYisgmkvPNLUj24QnMb5zlKyDiYjtTEXLiWWjxKJu/nFl4eQvACpyZxIBRYEw2bhmn5RohbLW/yFU8UqCROuRKjtygN4nNvhYnaAi1AbT3eT/3KpCfQM5QIqE2z5GyXQNvvMC62/4L+Crg6vxWn2pzikrJAnvcmBdkOakr94A5S0KARY6whTppoLm3msY/Pn9LX0DxZl+qRoChbBnOlWfM4aHqfaQD/PzMngmvSYvWiXfI0EsF5wS6zRyz8es61CDv9Yr0gef4CZ6WqTydCiZnikKyxcJOXrzccSAlnRHqdgQG6IN1FplsiRPRinljgBkIvY1yj+pRHt+sJmb+vewomjZFKgxlxIZFv2SyPzW/xR9cDYV7LqJGLWlprs8nJVL8s9C/JyDS30q59Dn8RzHHkwHvjVovrt9o/MCjCtVv3frDdvfVnxZZcRqW4PFwFu5ynmdz9/rct1jjRecvR1QIr/e1iqISaWQ/TnC5+Q+AiPpy6YSX+nRQdr1avRvN4YV4mo9LhFox7tW6Ke9DQgl0SqZtn0ffSSg2pl1wLQhJgdSw2waWwzE834O7rrpphhkfmPrRkhw4+0GMTB/k0HqsmBGQvjrKSkYQbGyNkz1dAIKYDYi1C6dS4Iu+nAZgCG+Kj6ZNb1rifzddfV8OKbU+SeFRoHql8XXPq6bNQlMm+36iNhn1ylMhEjcx4GmO/RLa4X29QPxx0GDfWu/Vong9ymSaD58xiBz8lrFG/GdFMb9MzcI=" } }, { "type": "step", "primary": "Write in the standard form $$a_{n}x^{n}+\\ldots\\:+a_{1}x+a_{0}=0$$", "result": "-16y^{4}-8y^{2}+3=0" }, { "type": "step", "primary": "Rewrite the equation with $$u=y^{2}$$ and $$u^{2}=y^{4}$$", "result": "-16u^{2}-8u+3=0" }, { "type": "interim", "title": "Solve $$-16u^{2}-8u+3=0:{\\quad}u=-\\frac{3}{4},\\:u=\\frac{1}{4}$$", "input": "-16u^{2}-8u+3=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-16u^{2}-8u+3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}}{2\\left(-16\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-16,\\:b=-8,\\:c=3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}}{2\\left(-16\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}=16$$", "input": "\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-8\\right)^{2}+4\\cdot\\:16\\cdot\\:3}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-8\\right)^{2}=8^{2}$$" ], "result": "=\\sqrt{8^{2}+4\\cdot\\:16\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:16\\cdot\\:3=192$$", "result": "=\\sqrt{8^{2}+192}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{64+192}" }, { "type": "step", "primary": "Add the numbers: $$64+192=256$$", "result": "=\\sqrt{256}" }, { "type": "step", "primary": "Factor the number: $$256=16^{2}$$", "result": "=\\sqrt{16^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{16^{2}}=16$$" ], "result": "=16", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cd7Mdcgz1T8BBymvdP1bZKZaDy8QprnC7F0Y6o9lNR18kR7hsO/rTOTBE0w4+r1R2P4wdaPUWsppya5Mp2DFUD/L0MoYg+CUn6oyL3EO7Yo2TrM2SYFEyYvN6nWs6MPcxDjiYmwbSQxDIiAX+RECwpg+EPZTeZAXqCmvZuv4qkM=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)},\\:{u}_{2}=\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)}:{\\quad}-\\frac{3}{4}$$", "input": "\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{8+16}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Add the numbers: $$8+16=24$$", "result": "=\\frac{24}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{24}{-32}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{24}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=-\\frac{3}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e4xoRT8wCESiOFOMMQ+D25wLY/Y4clRXQ6mGjyvXlsHNGoPE9TME3q+OPmgkv2RQgRqwxBWz3FT/O9/Ay4NEY1O1T0snFOPqKXL+S6MxVmnOnNKB/pT4hdTtksPjTrAoTH9Q4ZnPq9BPuFEDN0kgtyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}:{\\quad}\\frac{1}{4}$$", "input": "\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{8-16}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Subtract the numbers: $$8-16=-8$$", "result": "=\\frac{-8}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{-8}{-32}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{8}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75o8AVGA7tNYx1BfA+q+YSJwLY/Y4clRXQ6mGjyvXlsHNGoPE9TME3q+OPmgkv2RQiEw6G4T+RFI2ZfZDoB3kMvsicDtr1/4SZLlnwrW0smM0g6ajuMwUqvyouFMgKZ/idyHCJ8rDCeaGBfSEg5D1fg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{3}{4},\\:u=\\frac{1}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=-\\frac{3}{4},\\:u=\\frac{1}{4}" }, { "type": "step", "primary": "Substitute back $$u=y^{2},\\:$$solve for $$y$$" }, { "type": "interim", "title": "Solve $$y^{2}=-\\frac{3}{4}:{\\quad}$$No Solution for $$y\\in\\mathbb{R}$$", "input": "y^{2}=-\\frac{3}{4}", "steps": [ { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:y\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$y^{2}=\\frac{1}{4}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "y^{2}=\\frac{1}{4}", "steps": [ { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "y=\\sqrt{\\frac{1}{4}},\\:y=-\\sqrt{\\frac{1}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{1}{4}}=\\frac{1}{2}$$", "input": "\\sqrt{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{1}}{\\sqrt{4}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "secondary": [ "$$\\sqrt{1}=1$$" ], "result": "=\\frac{1}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0NzITgDFbnE8wkcXKMHdOwBSrju+5Z51e/ZZSD3gRHwjBZsqxqhl2a6oRKVJk8034tWRLd2VwIqlBNByF6663syTWcLcA3FbS+MZ1fFIklJt5MCuZPgBpwTTzu2tuLa/8abCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{1}{4}}=-\\frac{1}{2}$$", "input": "-\\sqrt{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=-\\frac{\\sqrt{1}}{\\sqrt{4}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "secondary": [ "$$\\sqrt{1}=1$$" ], "result": "=-\\frac{1}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+ArxmzTrBkvKbZrfJnwMGgx/VgUJQJZuTAY5js+oqjdT8kslx3FTmhk5oBBtojppJ/bq4/8//6/nV5O4fb8Xgwi7mapvsmMaNg8JzlNopDoeZ4sisPHFuuTLWzCcuewLnsue2m6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}y=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\left(-\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$4y=0:{\\quad}y=0$$", "input": "4y=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4y=0", "result": "y=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4y}{4}=\\frac{0}{4}" }, { "type": "step", "primary": "Simplify", "result": "y=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tYJ6987Pf3B1/fevMinHc3qmU5rQtJ8zdtqJ0kjhpCmAihI8YpOa7YjcZorvECwsjiXrEqQc/hA03woLzqUpjL5DAp1xNz5D3W6mLNf4cVxPoT5n6VgV9GoCdDbqwO+cLC1i/6Bg/vlotNlxPsrgnRtdrnyhOteu+4JV5WZhspZNkYfxKoe1Ra5hrZ2nDRu6pwbsDK6IpYzNzG9hhpG/yfrUh+mRFFUZO11QbyP/TQJXUOvJH4n8Qjg726uvijixJv3vPhyJEHOBe1ydiXxyKXBF/A4v44mfCY3wL9LQTkVPhtrepVIWql4Sa5xh9L5YV2cux1ConOHo1JnGykP6V0sXiOOw6wqajw39NwQqTA+meDhf1ouOhmSGkQIXxlV9E49ojGq96AXMwvFnC2BErN6vq8ch6QORd3MMFeFuyk/vXWUZGmN1xP57oyVC1eojckUUwA3eomiIEk/ipJ3VORvRZnb0zg9eVbQFTH50pQpXh3SFl4yPunBFcCOYFXDYDlBwOA1fZJXUaXJZN+u1UINmnreEEaGOyS+4N+t9v1n4zjvBTCh95txzZviGUMMJeB3JNIgAt4okc53Rulv5qSxd/3OQYxmsr5R6AryEWpbNtW+xt7YXAHnnUXfOhpt2FylN12FcHo+l2d9qBsS/3H5z9JkYomU9P1YGTFRFqDdN5Aod6Hr1Lp2e/29KhSgUYDSeuxDg/JbzQA94n1cRKI+UWTGJhgn934Rn/We++2w=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "y=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "step", "primary": "Plug the solutions $$y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$ into $$2xy=-\\frac{\\sqrt{3}}{2}$$" }, { "type": "interim", "title": "For $$2xy=-\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$\\frac{1}{2}:{\\quad}x=-\\frac{\\sqrt{3}}{2}$$", "steps": [ { "type": "step", "primary": "For $$2xy=-\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$\\frac{1}{2}$$", "result": "2x\\frac{1}{2}=-\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Solve $$2x\\frac{1}{2}=-\\frac{\\sqrt{3}}{2}:{\\quad}x=-\\frac{\\sqrt{3}}{2}$$", "input": "2x\\frac{1}{2}=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "\\frac{1\\cdot\\:2}{2}x=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "x\\cdot\\:1=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Multiply: $$x\\cdot\\:1=x$$", "result": "x=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "interim", "title": "For $$2xy=-\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$-\\frac{1}{2}:{\\quad}x=\\frac{\\sqrt{3}}{2}$$", "steps": [ { "type": "step", "primary": "For $$2xy=-\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$-\\frac{1}{2}$$", "result": "2x\\left(-\\frac{1}{2}\\right)=-\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Solve $$2x\\left(-\\frac{1}{2}\\right)=-\\frac{\\sqrt{3}}{2}:{\\quad}x=\\frac{\\sqrt{3}}{2}$$", "input": "2x\\left(-\\frac{1}{2}\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2\\left(-\\frac{1}{2}\\right)$$", "input": "2x\\left(-\\frac{1}{2}\\right)=-\\frac{\\sqrt{3}}{2}", "result": "x=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2\\left(-\\frac{1}{2}\\right)$$", "result": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}=\\frac{-\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}=\\frac{-\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}", "result": "x=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}:{\\quad}x$$", "input": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-2x\\frac{1}{2}}{-2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2x\\frac{1}{2}}{2\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "Multiply $$2x\\frac{1}{2}\\::{\\quad}x$$", "input": "2x\\frac{1}{2}", "result": "=\\frac{x}{2\\cdot\\:\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{1}{2}\\::{\\quad}1$$", "input": "2\\cdot\\:\\frac{1}{2}", "result": "=\\frac{x}{1}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UPIXVTEC4TES8SUzbtFWVBuWfJD+QqyHiGH9HMs0twyItVYK/n3VX9KvZ1p9/nRP3oZCq59Hq2va8/E5S/sf74xSWAquVNPtoc274CycqX6BBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2Se4S2rBuozrRmQpiqEPBsURiFJo2j5v1d4cqAt1Ub8wXxoSgZnAwPZwJgOMzZApZg==" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}:{\\quad}\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{-\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-\\frac{\\sqrt{3}}{2}}{-2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{\\frac{\\sqrt{3}}{2}}{2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{\\sqrt{3}}{2\\cdot\\:2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{3}}{4\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "Multiply $$4\\cdot\\:\\frac{1}{2}\\::{\\quad}2$$", "input": "4\\cdot\\:\\frac{1}{2}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVTVQn0Dd+FxZDRYQPY4AlXQaR468kzDUT2oW0Q13gptHfJEe4bDv60zkwRNMOPq9UULJUwysTkf+ifmThr1HGrqDz7IqY/3J3qrwMvKWPYN+Ec7ShOedm97LMngC0LVkYx4pgUWEah0lniZLlD4X0wvgd9/Fiv+sajCfJxgij2kXJyDS30q59Dn8RzHHkwHvjYsFfno0rtR6fii+PtVAR07kVgd/y5h80EeV6s7i7e5o" } }, { "type": "step", "result": "x=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z8cdVIYjtmNUeIiTNjbkt9PEUyKq7D9FBm9nszPGPrwnINLfSrn0OfxHMceTAe+NDlIPiilMMvZBQQ80sf2xV9NNksoGyucuJvjhDBT7x/gG8H0kr1D4S9K/1kWz425u0oynhwPSrER4ONMSexkysRMmrxIkf6FqT477xT6jMBgUur3A0DM0y/W0xb2BUxQhkw6d/GBgvPLki1Cdv904zQSCPcu2vYruuV0Hd6jcZXBzriYqA9DUajdRGodPS+H/l4gBAcaD25ntlCI3l1yi8Rc4KXs8xgJNiLkJwr0b/5PqkZQY+uh8n+ynJNeNS+3Zm34cBfDcaojJ3b5UEGrjn2k7sANVFNEZYaG+0BeCEq4Y9mzOC4MaF10FUqskBJFsLuye3cUxrA4ZtvAL6L/O2lAt3h6Q+0lpu50VSzy84yhiqbMrznmZIIhGCBTpaS4//K4PdyCD1yj4gzi0qFQsQnDtn+wtp0oxfyID0OIEJ+Wyq6317hZP5NJRJD3bfH2/gef4CZ6WqTydCiZnikKyxcJOXrzccSAlnRHqdgQG6INRdg+Kd+3u9ARt0Lbejxv5KrzidXXGd0hnmMrYJ1rVKgKdzFIw7m5jDzYWIu9TDjZt5aIO4/LAxIOJioZ8kIO4p+llO8m1JUsHsxYSyr5CjVb7qlKGJCpgSun0kdnDZYJH6n09x0sZXZjbeq1HH78kgsqwDvczEA2coxIljO6iORzS2phEsol5BFTVOAXt8l5V96z+yd/NTgi3X+wu1lkQ72wZm7kDUxdE6YSmfEbr2k2wYknsXP8G884wJnDhDuA9/ikVgcc14jkLo0ZkNPqUkzhOTQWSJK+8GDbWl55TFA==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "interim", "title": "Verify solutions by plugging them into the original equations", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$x^{2}-y^{2}=\\frac{1}{2}$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}:{\\quad}$$True", "input": "x^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}$$", "result": "\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}-\\left(-\\frac{1}{2}\\right)^{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{1}{2}=\\frac{1}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$x=-\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}:{\\quad}$$True", "input": "x^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=-\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}$$", "result": "\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}-\\left(\\frac{1}{2}\\right)^{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{1}{2}=\\frac{1}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "step", "primary": "Check the solutions by plugging them into $$2xy=-\\frac{\\sqrt{3}}{2}$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}:{\\quad}$$True", "input": "2xy=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}$$", "result": "2\\cdot\\:\\frac{\\sqrt{3}}{2}\\left(-\\frac{1}{2}\\right)=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Refine", "result": "-\\frac{\\sqrt{3}}{2}=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$x=-\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}:{\\quad}$$True", "input": "2xy=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=-\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}$$", "result": "2\\left(-\\frac{\\sqrt{3}}{2}\\right)\\frac{1}{2}=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Refine", "result": "-\\frac{\\sqrt{3}}{2}=-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface (many) 0Eq" } }, { "type": "step", "primary": "Therefore, the final solutions for $$x^{2}-y^{2}=\\frac{1}{2},\\:2xy=-\\frac{\\sqrt{3}}{2}$$ are ", "result": "\\begin{pmatrix}x=-\\frac{\\sqrt{3}}{2},\\:&y=\\frac{1}{2}\\\\x=\\frac{\\sqrt{3}}{2},\\:&y=-\\frac{1}{2}\\end{pmatrix}" } ], "meta": { "solvingClass": "System of Equations", "interimType": "Nonlinear Top 0Eq" } }, { "type": "step", "primary": "Substitute back $$u=x+yi$$", "result": "u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u^{2}=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}:{\\quad}u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i$$", "input": "u^{2}=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Substitute $$u=x+yi$$", "result": "\\left(x+yi\\right)^{2}=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Expand $$\\left(x+yi\\right)^{2}:{\\quad}\\left(x^{2}-y^{2}\\right)+2ixy$$", "input": "\\left(x+yi\\right)^{2}", "result": "\\left(x^{2}-y^{2}\\right)+2ixy=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "result": "=\\left(x+iy\\right)^{2}" }, { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=yi$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}+2xyi+\\left(yi\\right)^{2}" }, { "type": "interim", "title": "$$\\left(yi\\right)^{2}=-y^{2}$$", "input": "\\left(yi\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=i^{2}y^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$i^{2}=-1$$", "input": "i^{2}", "steps": [ { "type": "step", "primary": "Apply imaginary number rule: $$i^{2}=-1$$", "result": "=-1", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ixd6+oXBeBbBYSxiL8OAvAlAlm5MBjmOz6iqN1PySyUyzPuBwg0tEkeKHbqH8v7p1V4jc9gN7k5UVTYGDf0b7CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(-1\\right)y^{2}" }, { "type": "step", "primary": "Refine", "result": "=-y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76BIpiHUJqX+Gjz6Fwr0+9c0ag8T1MwTer44+aCS/ZFC+KWdrKsfvNYjtpYgUTWx00q1f0AvAEP8N09ISyDIZLcjMJz/TOsacQU4ZUGiOLLI=" } }, { "type": "step", "result": "=x^{2}+2ixy-y^{2}" }, { "type": "interim", "title": "Rewrite $$x^{2}+2ixy-y^{2}$$ in standard complex form: $$\\left(x^{2}-y^{2}\\right)+2xyi$$", "input": "x^{2}+2ixy-y^{2}", "steps": [ { "type": "step", "primary": "Group the real part and the imaginary part of the complex number", "result": "=\\left(x^{2}-y^{2}\\right)+2xyi" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fyAweyQvVwJsSQQKDjm97PG9YG39LAdblepQizswpHwa2CjokqvmBtYUm1DzxmMxKwNwzwWuFelyz8uvoD3mF2RLd2VwIqlBNByF6663sySYtHHZIJ8Nhz40VaXBzMWWxf2ToaH1kMvtQ3yodc/G8DRwLaryLWYBUQDrY2Dn8OshJnnwHxnMGhyjcgBVVR+ihbXRsjO6u6m4ZMlRD4wxtQ==" } }, { "type": "step", "result": "=\\left(x^{2}-y^{2}\\right)+2xyi" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7acBL/XWGdfMl4uhsCDRDUqr/LfvPjgKL/AHdgs1FR4p1DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhlGg7dnDDOhlC500PDgRQdo+i0Ux3lprvX50CFfl5rrAQ==" } }, { "type": "step", "primary": "Complex numbers can be equal only if their real and imaginary parts are equal", "secondary": [ "Rewrite as system of equations:" ], "result": "\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=\\frac{\\sqrt{3}}{2}\\end{bmatrix}" }, { "type": "interim", "title": "$$\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=\\frac{\\sqrt{3}}{2}\\end{bmatrix}:{\\quad}\\begin{pmatrix}x=\\frac{\\sqrt{3}}{2},\\:&y=\\frac{1}{2}\\\\x=-\\frac{\\sqrt{3}}{2},\\:&y=-\\frac{1}{2}\\end{pmatrix}$$", "input": "\\begin{bmatrix}x^{2}-y^{2}=\\frac{1}{2}\\\\2xy=\\frac{\\sqrt{3}}{2}\\end{bmatrix}", "steps": [ { "type": "interim", "title": "Isolate $$x\\:$$for $$2xy=\\frac{\\sqrt{3}}{2}:{\\quad}x=\\frac{\\sqrt{3}}{4y}$$", "input": "2xy=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2y$$", "input": "2xy=\\frac{\\sqrt{3}}{2}", "result": "x=\\frac{\\sqrt{3}}{4y}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2y$$", "result": "\\frac{2xy}{2y}=\\frac{\\frac{\\sqrt{3}}{2}}{2y}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2xy}{2y}=\\frac{\\frac{\\sqrt{3}}{2}}{2y}", "result": "x=\\frac{\\sqrt{3}}{4y}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2xy}{2y}:{\\quad}x$$", "input": "\\frac{2xy}{2y}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\frac{xy}{y}" }, { "type": "step", "primary": "Cancel the common factor: $$y$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aLZIDEtiNWvyIQ1h+ZnJcnyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz3vPC+dAdNgr+i2Xzs9Nk6DialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{\\sqrt{3}}{2}}{2y}:{\\quad}\\frac{\\sqrt{3}}{4y}$$", "input": "\\frac{\\frac{\\sqrt{3}}{2}}{2y}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{\\sqrt{3}}{2\\cdot\\:2y}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{3}}{4y}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajYh4gM/a8a9I1KqxtDo+lE1nny2DKlD8jWqJzs4vD4o2/aL2Coo0GISQwm8bv5wDicbGEQkNUVSmvjgyh8ffZpDC8bMazEaimkXU4rk8Wq2nmNwEgvmmajzdQwVlFLMmxS3xXPiuLkj+jJktEBj2/1KDz7IqY/3J3qrwMvKWPYN+eZ5+ud13iKeRtVq+iPI2gw==" } }, { "type": "step", "result": "x=\\frac{\\sqrt{3}}{4y}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "Plug the solutions $$x=\\frac{\\sqrt{3}}{4y}$$ into $$x^{2}-y^{2}=\\frac{1}{2}$$" }, { "type": "interim", "title": "For $$x^{2}-y^{2}=\\frac{1}{2}$$, subsitute $$x$$ with $$\\frac{\\sqrt{3}}{4y}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "steps": [ { "type": "step", "primary": "For $$x^{2}-y^{2}=\\frac{1}{2}$$, subsitute $$x$$ with $$\\frac{\\sqrt{3}}{4y}$$", "result": "\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}" }, { "type": "interim", "title": "Solve $$\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Multiply by LCM", "input": "\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}=\\frac{1}{2}", "result": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}:{\\quad}\\frac{3}{16y^{2}}$$", "input": "\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}", "result": "\\frac{3}{16y^{2}}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{\\left(4y\\right)^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(4y\\right)^{2}=4^{2}y^{2}$$" ], "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{4^{2}y^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3}{4^{2}y^{2}}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\frac{3}{16y^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7W+TF9DwfoU38VHY68TaD4+fxQujzrdfDjLDVNsnIkd3ehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgptQ0MC+bwOAymhr5hFfxeWLvbBmbuQNTF0TphKZ8Ruva1mNSL+EPUU5ZXrBhlxHaWCb97z4ciRBzgXtcnYl8cimGzB0JrX3MRULTZyKhG21i" } }, { "type": "interim", "title": "Find Least Common Multiplier of $$16y^{2},\\:2:{\\quad}16y^{2}$$", "input": "16y^{2},\\:2", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Least Common Multiplier of $$16,\\:2:{\\quad}16$$", "input": "16,\\:2", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$16:{\\quad}2\\cdot\\:2\\cdot\\:2\\cdot\\:2$$", "input": "16", "steps": [ { "type": "step", "primary": "$$16\\:$$divides by $$2\\quad\\:16=8\\cdot\\:2$$", "result": "=2\\cdot\\:8" }, { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdy5Ljmr36t3AP8UKimRmJwFB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECWYh2nZH9X0Rq5Oe4GdfGo" } }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$16$$ or $$2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:2\\cdot\\:2=16$$", "result": "=16" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$16y^{2}$$ or $$2$$", "result": "=16y^{2}" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top in Equation Title 1Eq" } }, { "type": "step", "primary": "Multiply by LCM=$$16y^{2}$$", "result": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}-y^{2}\\cdot\\:16y^{2}=\\frac{1}{2}\\cdot\\:16y^{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}-y^{2}\\cdot\\:16y^{2}=\\frac{1}{2}\\cdot\\:16y^{2}", "result": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3}{16y^{2}}\\cdot\\:16y^{2}:{\\quad}3$$", "input": "\\frac{3}{16y^{2}}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:16y^{2}}{16y^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$16$$", "result": "=\\frac{3y^{2}}{y^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$y^{2}$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iN4dvGcm+bQGP3fStnffSNsO8nDHBnMsuZu4AFRSIkBZ8mEc9fB7wfq6et5j4rXPgeXEFCD58rDFDfybZf16JD/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10YboD4oQjQoml9oHnDEg4Hb6n+FJ66//gnBohluY6VGj6LRTHeWmu9fnQIV+XmusB" } }, { "type": "interim", "title": "Simplify $$-y^{2}\\cdot\\:16y^{2}:{\\quad}-16y^{4}$$", "input": "-y^{2}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$y^{2}y^{2}=\\:y^{2+2}$$" ], "result": "=-16y^{2+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=-16y^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s708LaLmsXDBKqBgcrTGi8+m+2jZ3RVS4dK1SlxcsDTtWrju+5Z51e/ZZSD3gRHwjBCLyOc3zohAErAg63DYY3+4EFMST8lDZxn1Yq5HMKVTs6Zt5XsJOFJ/EboUEQ+/cI/hKRA7B3zygP4XVuJ6GHk48BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:16y^{2}:{\\quad}8y^{2}$$", "input": "\\frac{1}{2}\\cdot\\:16y^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:16}{2}y^{2}" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:16}{2}=8$$", "input": "\\frac{1\\cdot\\:16}{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:16=16$$", "result": "=\\frac{16}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{16}{2}=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCmUJCqRgq/EpL80lkhI7O+YJQJZuTAY5js+oqjdT8ksl2+Bm4xvhfGGRR5Jko7aCV/QVTV4d0XUrkjGhi3/crx0OteLnHolAnrbIwv7NbB+T" } }, { "type": "step", "result": "=8y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8crqNKUhefhKe0Zr7BxGuZQgJ/ZZA32ZInFBpDtxBfiKQdmmBkyA4nheYWVD5Bw8o2RLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9d4FJuPgO/OHNuxTBtnbIQjyxDSHcil1+wqic2arrKOM=" } }, { "type": "step", "result": "3-16y^{4}=8y^{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Equation LCM Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjACoC2ebXdWKUcWjG7mjemDg8+yKmP9yd6q8DLylj2DfjHiqCVW1qf/Tir0BXuG26CWTcnIoZeuC9MIszelWQR8ZUECKZNbPn9tzIDMOOYWXbv29AM9/HOqsgCGdzeTpfhkS3dlcCKpQTQcheuut7Mkcxxt1l9vi3NrYViHERcBSKzPY2SKV23dHfRIcwmcVAU=" } }, { "type": "interim", "title": "Solve $$3-16y^{4}=8y^{2}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "3-16y^{4}=8y^{2}", "steps": [ { "type": "interim", "title": "Move $$8y^{2}\\:$$to the left side", "input": "3-16y^{4}=8y^{2}", "result": "3-16y^{4}-8y^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$8y^{2}$$ from both sides", "result": "3-16y^{4}-8y^{2}=8y^{2}-8y^{2}" }, { "type": "step", "primary": "Simplify", "result": "3-16y^{4}-8y^{2}=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$a_{n}x^{n}+\\ldots\\:+a_{1}x+a_{0}=0$$", "result": "-16y^{4}-8y^{2}+3=0" }, { "type": "step", "primary": "Rewrite the equation with $$u=y^{2}$$ and $$u^{2}=y^{4}$$", "result": "-16u^{2}-8u+3=0" }, { "type": "interim", "title": "Solve $$-16u^{2}-8u+3=0:{\\quad}u=-\\frac{3}{4},\\:u=\\frac{1}{4}$$", "input": "-16u^{2}-8u+3=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-16u^{2}-8u+3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}}{2\\left(-16\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-16,\\:b=-8,\\:c=3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}}{2\\left(-16\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EGm9Nk0nwq0YSBApvSlOsUDqE5YOSTSQY6sLd+1n7SbUVDWobp8m5aHXAod5ZVNlXpujYDTIG5+uylY5s5s7l7TACF0rqSUCZYic0Tyt9ZMxi9jKbtD+E/BBby3MO11PJsj7/Kk0U1YAzG96pLnGpcWGLS0U7sT1xuwCq90Wt19UVaYyEbxHgQSXt4Bzyw7C6dDnzgLQ7aMLG/urg9ikYE+hPmfpWBX0agJ0NurA75wsLWL/oGD++Wi02XE+yuCdG12ufKE61677glXlZmGylk2Rh/Eqh7VFrmGtnacNG7qnBuwMroiljM3Mb2GGkb/J+tSH6ZEUVRk7XVBvI/9NAldQ68kfifxCODvbq6+KOLEm/e8+HIkQc4F7XJ2JfHIpcEX8Di/jiZ8JjfAv0tBORU+G2t6lUhaqXhJrnGH0vlhXZy7HUKic4ejUmcbKQ/pXSxeI47DrCpqPDf03BCpMD6Z4OF/Wi46GZIaRAhfGVX0Tj2iMar3oBczC8WcLYESs3q+rxyHpA5F3cwwV4W7KT+9dZRkaY3XE/nujJULV6iNyRRTADd6iaIgST+KkndU5G9FmdvTOD15VtAVMfnSlCleHdIWXjI+6cEVwI5gVcNgOUHA4DV9kldRpclk367VQg2aet4QRoY7JL7g3632/WfjOO8FMKH3m3HNm+IZQwwl4Hck0iAC3iiRzndG6W/mpLF3/c5BjGayvlHoCvIRals21b7G3thcAeedRd86Gm3YXKU3XYVwej6XZ32oGxL/cfnP0mRiiZT0/VgZMVEWoN03kCh3oevUunZ7/b0qFKBQV2XdfGQazN5+PiGoJEg3SanoZPShdwJWDv/wN7YGbQImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}=16$$", "input": "\\sqrt{\\left(-8\\right)^{2}-4\\left(-16\\right)\\cdot\\:3}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-8\\right)^{2}+4\\cdot\\:16\\cdot\\:3}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-8\\right)^{2}=8^{2}$$" ], "result": "=\\sqrt{8^{2}+4\\cdot\\:16\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:16\\cdot\\:3=192$$", "result": "=\\sqrt{8^{2}+192}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{64+192}" }, { "type": "step", "primary": "Add the numbers: $$64+192=256$$", "result": "=\\sqrt{256}" }, { "type": "step", "primary": "Factor the number: $$256=16^{2}$$", "result": "=\\sqrt{16^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{16^{2}}=16$$" ], "result": "=16", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cd7Mdcgz1T8BBymvdP1bZKZaDy8QprnC7F0Y6o9lNR18kR7hsO/rTOTBE0w4+r1R2P4wdaPUWsppya5Mp2DFUD/L0MoYg+CUn6oyL3EO7Yo2TrM2SYFEyYvN6nWs6MPcxDjiYmwbSQxDIiAX+RECwpg+EPZTeZAXqCmvZuv4qkM=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)},\\:{u}_{2}=\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)}:{\\quad}-\\frac{3}{4}$$", "input": "\\frac{-\\left(-8\\right)+16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{8+16}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Add the numbers: $$8+16=24$$", "result": "=\\frac{24}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{24}{-32}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{24}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=-\\frac{3}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e4xoRT8wCESiOFOMMQ+D25wLY/Y4clRXQ6mGjyvXlsHNGoPE9TME3q+OPmgkv2RQgRqwxBWz3FT/O9/Ay4NEY1O1T0snFOPqKXL+S6MxVmnOnNKB/pT4hdTtksPjTrAoTH9Q4ZnPq9BPuFEDN0kgtyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}:{\\quad}\\frac{1}{4}$$", "input": "\\frac{-\\left(-8\\right)-16}{2\\left(-16\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{8-16}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Subtract the numbers: $$8-16=-8$$", "result": "=\\frac{-8}{-2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{-8}{-32}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{8}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75o8AVGA7tNYx1BfA+q+YSJwLY/Y4clRXQ6mGjyvXlsHNGoPE9TME3q+OPmgkv2RQiEw6G4T+RFI2ZfZDoB3kMvsicDtr1/4SZLlnwrW0smM0g6ajuMwUqvyouFMgKZ/idyHCJ8rDCeaGBfSEg5D1fg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{3}{4},\\:u=\\frac{1}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=-\\frac{3}{4},\\:u=\\frac{1}{4}" }, { "type": "step", "primary": "Substitute back $$u=y^{2},\\:$$solve for $$y$$" }, { "type": "interim", "title": "Solve $$y^{2}=-\\frac{3}{4}:{\\quad}$$No Solution for $$y\\in\\mathbb{R}$$", "input": "y^{2}=-\\frac{3}{4}", "steps": [ { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:y\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$y^{2}=\\frac{1}{4}:{\\quad}y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$", "input": "y^{2}=\\frac{1}{4}", "steps": [ { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "y=\\sqrt{\\frac{1}{4}},\\:y=-\\sqrt{\\frac{1}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{1}{4}}=\\frac{1}{2}$$", "input": "\\sqrt{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{1}}{\\sqrt{4}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "secondary": [ "$$\\sqrt{1}=1$$" ], "result": "=\\frac{1}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0NzITgDFbnE8wkcXKMHdOwBSrju+5Z51e/ZZSD3gRHwjBZsqxqhl2a6oRKVJk8034tWRLd2VwIqlBNByF6663syTWcLcA3FbS+MZ1fFIklJt5MCuZPgBpwTTzu2tuLa/8abCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{1}{4}}=-\\frac{1}{2}$$", "input": "-\\sqrt{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=-\\frac{\\sqrt{1}}{\\sqrt{4}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "secondary": [ "$$\\sqrt{1}=1$$" ], "result": "=-\\frac{1}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+ArxmzTrBkvKbZrfJnwMGgx/VgUJQJZuTAY5js+oqjdT8kslx3FTmhk5oBBtojppJ/bq4/8//6/nV5O4fb8Xgwi7mapvsmMaNg8JzlNopDoeZ4sisPHFuuTLWzCcuewLnsue2m6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}y=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\left(\\frac{\\sqrt{3}}{4y}\\right)^{2}-y^{2}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$4y=0:{\\quad}y=0$$", "input": "4y=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4y=0", "result": "y=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4y}{4}=\\frac{0}{4}" }, { "type": "step", "primary": "Simplify", "result": "y=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "y=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "y=\\frac{1}{2},\\:y=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "step", "primary": "Plug the solutions $$y=\\frac{1}{2},\\:y=-\\frac{1}{2}$$ into $$2xy=\\frac{\\sqrt{3}}{2}$$" }, { "type": "interim", "title": "For $$2xy=\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$\\frac{1}{2}:{\\quad}x=\\frac{\\sqrt{3}}{2}$$", "steps": [ { "type": "step", "primary": "For $$2xy=\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$\\frac{1}{2}$$", "result": "2x\\frac{1}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Solve $$2x\\frac{1}{2}=\\frac{\\sqrt{3}}{2}:{\\quad}x=\\frac{\\sqrt{3}}{2}$$", "input": "2x\\frac{1}{2}=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "\\frac{1\\cdot\\:2}{2}x=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "x\\cdot\\:1=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Multiply: $$x\\cdot\\:1=x$$", "result": "x=\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "interim", "title": "For $$2xy=\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$-\\frac{1}{2}:{\\quad}x=-\\frac{\\sqrt{3}}{2}$$", "steps": [ { "type": "step", "primary": "For $$2xy=\\frac{\\sqrt{3}}{2}$$, subsitute $$y$$ with $$-\\frac{1}{2}$$", "result": "2x\\left(-\\frac{1}{2}\\right)=\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Solve $$2x\\left(-\\frac{1}{2}\\right)=\\frac{\\sqrt{3}}{2}:{\\quad}x=-\\frac{\\sqrt{3}}{2}$$", "input": "2x\\left(-\\frac{1}{2}\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2\\left(-\\frac{1}{2}\\right)$$", "input": "2x\\left(-\\frac{1}{2}\\right)=\\frac{\\sqrt{3}}{2}", "result": "x=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2\\left(-\\frac{1}{2}\\right)$$", "result": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}=\\frac{\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}=\\frac{\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}", "result": "x=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}:{\\quad}x$$", "input": "\\frac{2x\\left(-\\frac{1}{2}\\right)}{2\\left(-\\frac{1}{2}\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-2x\\frac{1}{2}}{-2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2x\\frac{1}{2}}{2\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "Multiply $$2x\\frac{1}{2}\\::{\\quad}x$$", "input": "2x\\frac{1}{2}", "result": "=\\frac{x}{2\\cdot\\:\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{1}{2}\\::{\\quad}1$$", "input": "2\\cdot\\:\\frac{1}{2}", "result": "=\\frac{x}{1}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UPIXVTEC4TES8SUzbtFWVBuWfJD+QqyHiGH9HMs0twyItVYK/n3VX9KvZ1p9/nRP3oZCq59Hq2va8/E5S/sf74xSWAquVNPtoc274CycqX6BBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2Se4S2rBuozrRmQpiqEPBsURiFJo2j5v1d4cqAt1Ub8wXxoSgZnAwPZwJgOMzZApZg==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}:{\\quad}-\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{\\frac{\\sqrt{3}}{2}}{2\\left(-\\frac{1}{2}\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{\\frac{\\sqrt{3}}{2}}{-2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{\\sqrt{3}}{2}}{2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{\\sqrt{3}}{2}}{2\\cdot\\:\\frac{1}{2}}=\\frac{\\sqrt{3}}{2\\cdot\\:2\\cdot\\:\\frac{1}{2}}$$" ], "result": "=-\\frac{\\sqrt{3}}{2\\cdot\\:2\\cdot\\:\\frac{1}{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=-\\frac{\\sqrt{3}}{4\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "Multiply $$4\\cdot\\:\\frac{1}{2}\\::{\\quad}2$$", "input": "4\\cdot\\:\\frac{1}{2}", "result": "=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajYh4gM/a8a9I1KqxtDo+lE3HEVwBh8xi1kBDTg0hlUcZAJYpRu9XpYrd8NSAW2DdD5W7TL+fKYUW9qkW83QPn7yDz7IqY/3J3qrwMvKWPYN+Ec7ShOedm97LMngC0LVkYx4pgUWEah0lniZLlD4X0wtEEedfW/3lEhzme5c9xQJX3q+rxyHpA5F3cwwV4W7KT6r67wJlpUz3T9687BujLZikI+jSotxqO7F/TSXuhSqN" } }, { "type": "step", "result": "x=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z8cdVIYjtmNUeIiTNjbktyDC9ckRVQySXdNTiP9ty4Ter6vHIekDkXdzDBXhbspPHI5S0StY1FdtOqqOPr0Te4M6C8XPkFBvtqZsfoTImIxbHafqHCSsRspOfun7h8PJmnKogegErdht+ZMUrN+ZuWKRu+/57kEw4g+NGoP4+G8Uur3A0DM0y/W0xb2BUxQhkw6d/GBgvPLki1Cdv904zQSCPcu2vYruuV0Hd6jcZXBzriYqA9DUajdRGodPS+H/l4gBAcaD25ntlCI3l1yi8Rc4KXs8xgJNiLkJwr0b/5PqkZQY+uh8n+ynJNeNS+3Zm34cBfDcaojJ3b5UEGrjn2k7sANVFNEZYaG+0BeCEq4Y9mzOC4MaF10FUqskBJFsLuye3cUxrA4ZtvAL6L/O2lAt3h6Q+0lpu50VSzy84yhiqbMrznmZIIhGCBTpaS4//K4PdyCD1yj4gzi0qFQsQnDtn+wtp0oxfyID0OIEJ+Wyq6317hZP5NJRJD3bfH2/gef4CZ6WqTydCiZnikKyxcJOXrzccSAlnRHqdgQG6INRdg+Kd+3u9ARt0Lbejxv5KrzidXXGd0hnmMrYJ1rVKgKdzFIw7m5jDzYWIu9TDjZt5aIO4/LAxIOJioZ8kIO4p+llO8m1JUsHsxYSyr5CjVb7qlKGJCpgSun0kdnDZYJH6n09x0sZXZjbeq1HH78kgsqwDvczEA2coxIljO6iORzS2phEsol5BFTVOAXt8l5V96z+yd/NTgi3X+wu1lkQ72wZm7kDUxdE6YSmfEbr2k2wYknsXP8G884wJnDhDuA9/ikVgcc14jkLo0ZkNPqUkzhOTQWSJK+8GDbWl55TFA==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Generic Substitute Specific 3Eq" } }, { "type": "interim", "title": "Verify solutions by plugging them into the original equations", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$x^{2}-y^{2}=\\frac{1}{2}$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=-\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}:{\\quad}$$True", "input": "x^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=-\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}$$", "result": "\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}-\\left(-\\frac{1}{2}\\right)^{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{1}{2}=\\frac{1}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$x=\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}:{\\quad}$$True", "input": "x^{2}-y^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}$$", "result": "\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}-\\left(\\frac{1}{2}\\right)^{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{1}{2}=\\frac{1}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "step", "primary": "Check the solutions by plugging them into $$2xy=\\frac{\\sqrt{3}}{2}$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$x=-\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}:{\\quad}$$True", "input": "2xy=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=-\\frac{\\sqrt{3}}{2},\\:y=-\\frac{1}{2}$$", "result": "2\\left(-\\frac{\\sqrt{3}}{2}\\right)\\left(-\\frac{1}{2}\\right)=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$x=\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}:{\\quad}$$True", "input": "2xy=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Plug in $$x=\\frac{\\sqrt{3}}{2},\\:y=\\frac{1}{2}$$", "result": "2\\cdot\\:\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{1}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Refine", "result": "\\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface (many) 0Eq" } }, { "type": "step", "primary": "Therefore, the final solutions for $$x^{2}-y^{2}=\\frac{1}{2},\\:2xy=\\frac{\\sqrt{3}}{2}$$ are ", "result": "\\begin{pmatrix}x=\\frac{\\sqrt{3}}{2},\\:&y=\\frac{1}{2}\\\\x=-\\frac{\\sqrt{3}}{2},\\:&y=-\\frac{1}{2}\\end{pmatrix}" } ], "meta": { "solvingClass": "System of Equations", "interimType": "Nonlinear Top 0Eq" } }, { "type": "step", "primary": "Substitute back $$u=x+yi$$", "result": "u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:u=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:u=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(3θ\\right)$$", "result": "\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i,\\:\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(3θ\\right)=\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(3θ\\right)=-\\frac{\\sqrt{3}}{2}-\\frac{1}{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "\\mathrm{No\\:Solution\\:for}\\:θ\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sec^{2}(3θ)\\cos^{2}(3θ)-\\sec^{2}(3θ)-\\cos^{2}(3θ)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }