5r^2-12r+6=2

{ "query": { "display": "$$5r^{2}-12r+6=2$$", "symbolab_question": "EQUATION#5r^{2}-12r+6=2" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "r=2,r=\\frac{2}{5}", "decimal": "r=2,r=0.4", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$5r^{2}-12r+6=2$$", "symbolab_question": "quadraticformula 5r^{2}-12r+6=2" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$5r^{2}-12r+6=2$$", "symbolab_question": "completesquare 5r^{2}-12r+6=2" } }, { "method": "Solve by factoring", "query": { "display": "factor $$5r^{2}-12r+6=2$$", "symbolab_question": "factor 5r^{2}-12r+6=2" } } ], "steps": { "type": "interim", "title": "$$5r^{2}-12r+6=2{\\quad:\\quad}r=2,\\:r=\\frac{2}{5}$$", "input": "5r^{2}-12r+6=2", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the left side", "input": "5r^{2}-12r+6=2", "result": "5r^{2}-12r+4=0", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "5r^{2}-12r+6-2=2-2" }, { "type": "step", "primary": "Simplify", "result": "5r^{2}-12r+4=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "5r^{2}-12r+4=0", "result": "{r}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\cdot\\:5\\cdot\\:4}}{2\\cdot\\:5}", "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 $${\\quad}a=5,\\:b=-12,\\:c=4$$", "result": "{r}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\cdot\\:5\\cdot\\:4}}{2\\cdot\\:5}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-12\\right)^{2}-4\\cdot\\:5\\cdot\\:4}=8$$", "input": "\\sqrt{\\left(-12\\right)^{2}-4\\cdot\\:5\\cdot\\:4}", "result": "{r}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-12\\right)^{2}=12^{2}$$" ], "result": "=\\sqrt{12^{2}-4\\cdot\\:5\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:5\\cdot\\:4=80$$", "result": "=\\sqrt{12^{2}-80}" }, { "type": "step", "primary": "$$12^{2}=144$$", "result": "=\\sqrt{144-80}" }, { "type": "step", "primary": "Subtract the numbers: $$144-80=64$$", "result": "=\\sqrt{64}" }, { "type": "step", "primary": "Factor the number: $$64=8^{2}$$", "result": "=\\sqrt{8^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{8^{2}}=8$$" ], "result": "=8", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fr9oMXl8QtE4/ZTpPJTfW0aaUOeO/v1nJrgKI3GP3u9OXjDqgZceJmQvXbNh57qqcJChiVhDxT5N/LHSTLMjyN9SR5kmQ9WWEGRsYUUwojugOO7zcOuZPMrm94EVRKlkHgxSgNcpBF9iN9FoaULpE/Ykl86IaR8AfiwoxhmOt8U=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{r}_{1}=\\frac{-\\left(-12\\right)+8}{2\\cdot\\:5},\\:{r}_{2}=\\frac{-\\left(-12\\right)-8}{2\\cdot\\:5}" }, { "type": "interim", "title": "$$r=\\frac{-\\left(-12\\right)+8}{2\\cdot\\:5}:{\\quad}2$$", "input": "\\frac{-\\left(-12\\right)+8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{12+8}{2\\cdot\\:5}" }, { "type": "step", "primary": "Add the numbers: $$12+8=20$$", "result": "=\\frac{20}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{20}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{20}{10}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7anucnun7OX2G4yim0upPTsRe1RJVygxtUwjTv6xH4SbehkKrn0era9rz8TlL+x/vYS2YS91TkQh9pmLGKRo5TYqgR1r4wI+OUQBjUwRJBq2UGDFX+0geEpCWvkGy0k33ELV8YYBAEn1TDxvnBI6vUQ==" } }, { "type": "interim", "title": "$$r=\\frac{-\\left(-12\\right)-8}{2\\cdot\\:5}:{\\quad}\\frac{2}{5}$$", "input": "\\frac{-\\left(-12\\right)-8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{12-8}{2\\cdot\\:5}" }, { "type": "step", "primary": "Subtract the numbers: $$12-8=4$$", "result": "=\\frac{4}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{4}{10}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{2}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7E4vVNnhePDnnZC3zSqgvwsRe1RJVygxtUwjTv6xH4SbehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpkG68WxT//G2NSJzVXAyeefbCKye2MFrj35ktPJd7ZLBPZN6/diTBWu0sLtmpAq39xtMXMg3gbDKx5EjNEvR0Go=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "r=2,\\:r=\\frac{2}{5}" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "r", "plotRequest": "5r^{2}-12r+6-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }