2n^2-4=2n

{ "query": { "display": "$$2n^{2}-4=2n$$", "symbolab_question": "EQUATION#2n^{2}-4=2n" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "n=2,n=-1", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$2n^{2}-4=2n$$", "symbolab_question": "quadraticformula 2n^{2}-4=2n" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$2n^{2}-4=2n$$", "symbolab_question": "completesquare 2n^{2}-4=2n" } }, { "method": "Solve by factoring", "query": { "display": "factor $$2n^{2}-4=2n$$", "symbolab_question": "factor 2n^{2}-4=2n" } } ], "steps": { "type": "interim", "title": "$$2n^{2}-4=2n{\\quad:\\quad}n=2,\\:n=-1$$", "input": "2n^{2}-4=2n", "steps": [ { "type": "interim", "title": "Move $$2n\\:$$to the left side", "input": "2n^{2}-4=2n", "result": "2n^{2}-4-2n=0", "steps": [ { "type": "step", "primary": "Subtract $$2n$$ from both sides", "result": "2n^{2}-4-2n=2n-2n" }, { "type": "step", "primary": "Simplify", "result": "2n^{2}-4-2n=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "2n^{2}-2n-4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2n^{2}-2n-4=0", "result": "{n}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\left(-4\\right)}}{2\\cdot\\:2}", "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 $${\\quad}a=2,\\:b=-2,\\:c=-4$$", "result": "{n}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\left(-4\\right)}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\left(-4\\right)}=6$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\left(-4\\right)}", "result": "{n}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:6}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:2\\cdot\\:4}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}+4\\cdot\\:2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:4=32$$", "result": "=\\sqrt{2^{2}+32}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+32}" }, { "type": "step", "primary": "Add the numbers: $$4+32=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTm10PMZXbwJNrOCkHnzS1X0AlilG71elit3w1IBbYN0POfzEJvy01qFtiDDf1FBy4KN6Hv6MoTMtvtU0IQwXdn+yrEFCkHLDWs6zaAkN1X/8vIA+obKXFQu1EDfoJ2LQVCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{n}_{1}=\\frac{-\\left(-2\\right)+6}{2\\cdot\\:2},\\:{n}_{2}=\\frac{-\\left(-2\\right)-6}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$n=\\frac{-\\left(-2\\right)+6}{2\\cdot\\:2}:{\\quad}2$$", "input": "\\frac{-\\left(-2\\right)+6}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+6}{2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\frac{8}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{8}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFqNWcG7GsnnezBHiIMWJJKet2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUOaMMFIX2mms0JRLfYONHN+js3X5Ur8lPZHo6jPummXmx0ij68Lvr+79BVqXMQAj8w==" } }, { "type": "interim", "title": "$$n=\\frac{-\\left(-2\\right)-6}{2\\cdot\\:2}:{\\quad}-1$$", "input": "\\frac{-\\left(-2\\right)-6}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-6}{2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$2-6=-4$$", "result": "=\\frac{-4}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-4}{4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72LopWbBMxyRtD0Gow7zUspKet2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKSECk6GWu1Zs/UVqyB86KAh+MWrT350THcGkvbP3uZQ+H6+RpQztJ+C90UDS/CS+ffHoXIQE40FG2d5xHqpEqgg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "n=2,\\:n=-1" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2n^{2}-4-2n" }, "showViewLarger": true } }, "meta": { "showVerify": true } }