r^2-6r+2=0

{ "query": { "display": "$$r^{2}-6r+2=0$$", "symbolab_question": "EQUATION#r^{2}-6r+2=0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "r=3+\\sqrt{7},r=3-\\sqrt{7}", "decimal": "r=5.64575…,r=0.35424…", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$r^{2}-6r+2=0$$", "symbolab_question": "quadraticformula r^{2}-6r+2=0" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$r^{2}-6r+2=0$$", "symbolab_question": "completesquare r^{2}-6r+2=0" } } ], "steps": { "type": "interim", "title": "$$r^{2}-6r+2=0{\\quad:\\quad}r=3+\\sqrt{7},\\:r=3-\\sqrt{7}$$", "input": "r^{2}-6r+2=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "r^{2}-6r+2=0", "result": "{r}_{1,\\:2}=\\frac{-\\left(-6\\right)\\pm\\:\\sqrt{\\left(-6\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{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 $${\\quad}a=1,\\:b=-6,\\:c=2$$", "result": "{r}_{1,\\:2}=\\frac{-\\left(-6\\right)\\pm\\:\\sqrt{\\left(-6\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-6\\right)^{2}-4\\cdot\\:1\\cdot\\:2}=2\\sqrt{7}$$", "input": "\\sqrt{\\left(-6\\right)^{2}-4\\cdot\\:1\\cdot\\:2}", "result": "{r}_{1,\\:2}=\\frac{-\\left(-6\\right)\\pm\\:2\\sqrt{7}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-6\\right)^{2}=6^{2}$$" ], "result": "=\\sqrt{6^{2}-4\\cdot\\:1\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:2=8$$", "result": "=\\sqrt{6^{2}-8}" }, { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-8}" }, { "type": "step", "primary": "Subtract the numbers: $$36-8=28$$", "result": "=\\sqrt{28}" }, { "type": "interim", "title": "Prime factorization of $$28:{\\quad}2^{2}\\cdot\\:7$$", "input": "28", "result": "=\\sqrt{2^{2}\\cdot\\:7}", "steps": [ { "type": "step", "primary": "$$28\\:$$divides by $$2\\quad\\:28=14\\cdot\\:2$$", "result": "=2\\cdot\\:14" }, { "type": "step", "primary": "$$14\\:$$divides by $$2\\quad\\:14=7\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:7" }, { "type": "step", "primary": "$$2,\\:7$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:7" }, { "type": "step", "result": "=2^{2}\\cdot\\:7" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFyuEeNBgSa3LrIvx33A/jwUB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEBVROt24Xm+S5pVSuEVx9+J" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{7}\\sqrt{2^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{7}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7biV9ml2SRpXxmTVDdibQ9HGW/jTwUrqHAcqr1cHDsBBw4ba/3KCEGXsM04IlGc/HzMFYmi1F5Hg/ibpEToVnYzJOgsfENLKvcUpxDcFJRG+7QkeXSI8UOQEoPmTxVUrHoZavaOrrWo0P8l0H0zf3xjxSsBe/9AdtGHv5Ln8nZTXu4AwZUf8FfNLd/jU1dI7s" } }, { "type": "step", "primary": "Separate the solutions", "result": "{r}_{1}=\\frac{-\\left(-6\\right)+2\\sqrt{7}}{2\\cdot\\:1},\\:{r}_{2}=\\frac{-\\left(-6\\right)-2\\sqrt{7}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$r=\\frac{-\\left(-6\\right)+2\\sqrt{7}}{2\\cdot\\:1}:{\\quad}3+\\sqrt{7}$$", "input": "\\frac{-\\left(-6\\right)+2\\sqrt{7}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{6+2\\sqrt{7}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{6+2\\sqrt{7}}{2}" }, { "type": "interim", "title": "Factor $$6+2\\sqrt{7}:{\\quad}2\\left(3+\\sqrt{7}\\right)$$", "input": "6+2\\sqrt{7}", "result": "=\\frac{2\\left(3+\\sqrt{7}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:3+2\\sqrt{7}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(3+\\sqrt{7}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=3+\\sqrt{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zgiY4HVozNzOuV5q2rxgOgyAw/kFEQQMg9HPzhDUf5u2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyELGIezvPiQrwAPcUrAg8A+vJAECyBNoee/07JkEc7tSvNTOJSP1RY2Yvy0SEAcHP6U7XRdqRDPxH6tbFQ1lCEpsZSEP36ANCDTv0XrTRe7l" } }, { "type": "interim", "title": "$$r=\\frac{-\\left(-6\\right)-2\\sqrt{7}}{2\\cdot\\:1}:{\\quad}3-\\sqrt{7}$$", "input": "\\frac{-\\left(-6\\right)-2\\sqrt{7}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{6-2\\sqrt{7}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{6-2\\sqrt{7}}{2}" }, { "type": "interim", "title": "Factor $$6-2\\sqrt{7}:{\\quad}2\\left(3-\\sqrt{7}\\right)$$", "input": "6-2\\sqrt{7}", "result": "=\\frac{2\\left(3-\\sqrt{7}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:3-2\\sqrt{7}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(3-\\sqrt{7}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=3-\\sqrt{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HgaAmqs0AKuOxt5YH5MAqAyAw/kFEQQMg9HPzhDUf5u2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyINdQrDSiyroOx25+TY14oKvJAECyBNoee/07JkEc7tSC1sSOogGYFf1OrEP88Nz3qU7XRdqRDPxH6tbFQ1lCEpsZSEP36ANCDTv0XrTRe7l" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "r=3+\\sqrt{7},\\:r=3-\\sqrt{7}" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "r", "plotRequest": "r^{2}-6r+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }