x^2+12x-64=0

{ "query": { "display": "$$x^{2}+12x-64=0$$", "symbolab_question": "EQUATION#x^{2}+12x-64=0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "x=4,x=-16", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$x^{2}+12x-64=0$$", "symbolab_question": "quadraticformula x^{2}+12x-64=0" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$x^{2}+12x-64=0$$", "symbolab_question": "completesquare x^{2}+12x-64=0" } }, { "method": "Solve by factoring", "query": { "display": "factor $$x^{2}+12x-64=0$$", "symbolab_question": "factor x^{2}+12x-64=0" } } ], "steps": { "type": "interim", "title": "$$x^{2}+12x-64=0{\\quad:\\quad}x=4,\\:x=-16$$", "input": "x^{2}+12x-64=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}+12x-64=0", "result": "{x}_{1,\\:2}=\\frac{-12\\pm\\:\\sqrt{12^{2}-4\\cdot\\:1\\cdot\\:\\left(-64\\right)}}{2\\cdot\\:1}", "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 $${\\quad}a=1,\\:b=12,\\:c=-64$$", "result": "{x}_{1,\\:2}=\\frac{-12\\pm\\:\\sqrt{12^{2}-4\\cdot\\:1\\cdot\\:\\left(-64\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{12^{2}-4\\cdot\\:1\\cdot\\:\\left(-64\\right)}=20$$", "input": "\\sqrt{12^{2}-4\\cdot\\:1\\cdot\\:\\left(-64\\right)}", "result": "{x}_{1,\\:2}=\\frac{-12\\pm\\:20}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{12^{2}+4\\cdot\\:1\\cdot\\:64}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:64=256$$", "result": "=\\sqrt{12^{2}+256}" }, { "type": "step", "primary": "$$12^{2}=144$$", "result": "=\\sqrt{144+256}" }, { "type": "step", "primary": "Add the numbers: $$144+256=400$$", "result": "=\\sqrt{400}" }, { "type": "step", "primary": "Factor the number: $$400=20^{2}$$", "result": "=\\sqrt{20^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{20^{2}}=20$$" ], "result": "=20", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UIKTqwzY8/jQt4wJ26+eRUyltbx1ORuo8uWCi9bAKvi1Sd3MGjwhOQqC2WrEVQgJq47vuWedXv2WUg94ER8IwUjMXvW7eD1/XdlBXbtmwf6tKZuEUwOexGgqUsh8RS+lQWPsKI7wVA0NK3Y3PWacXtqlx+d4ftC80gA8Iz5EnA0=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-12+20}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-12-20}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-12+20}{2\\cdot\\:1}:{\\quad}4$$", "input": "\\frac{-12+20}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-12+20=8$$", "result": "=\\frac{8}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{8}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{2}=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yQSSeU/PSVLSg2F288b4uvebyPAEKYSzXI5q/94Em4ndd47a0hQ8flDbGsI5To1dlXydCAnPpeOlbdfKkqvrL6KrdhGZc6NOECaxJSgcKg4k/C3UCjO97roGJOOmVmkHmljozu9NUZjsVv59ndXDvA==" } }, { "type": "interim", "title": "$$x=\\frac{-12-20}{2\\cdot\\:1}:{\\quad}-16$$", "input": "\\frac{-12-20}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-12-20=-32$$", "result": "=\\frac{-32}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-32}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{32}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{32}{2}=16$$", "result": "=-16" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DuWyG9UcFEzgeXGxeGAPxfebyPAEKYSzXI5q/94Em4ndd47a0hQ8flDbGsI5To1dD4+U9Lm1lkxgR4PDVW/NphYNkJYllbXYspDtfT4FNOJ1rzfgjvUR16UCOPrcqBSS1FYlXu7xkLLsgM93YHqktw==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=4,\\:x=-16" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}+12x-64" }, "showViewLarger": true } }, "meta": { "showVerify": true } }