X^2=-4-4600*0.1*(X+2)

{ "query": { "display": "$$X^{2}=-4-4600\\cdot\\:0.1\\cdot\\:\\left(X+2\\right)$$", "symbolab_question": "EQUATION#X^{2}=-4-4600\\cdot 0.1\\cdot (X+2)" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "X=2(\\sqrt{12994}-115),X=-2(115+\\sqrt{12994})", "decimal": "X=-2.01754…,X=-457.98245…", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$X^{2}=-4-4600\\cdot\\:0.1\\cdot\\:\\left(X+2\\right)$$", "symbolab_question": "quadraticformula X^{2}=-4-4600\\cdot 0.1\\cdot (X+2)" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$X^{2}=-4-4600\\cdot\\:0.1\\cdot\\:\\left(X+2\\right)$$", "symbolab_question": "completesquare X^{2}=-4-4600\\cdot 0.1\\cdot (X+2)" } } ], "steps": { "type": "interim", "title": "$$X^{2}=-4-4600\\cdot\\:0.1\\left(X+2\\right){\\quad:\\quad}X=2\\left(\\sqrt{12994}-115\\right),\\:X=-2\\left(115+\\sqrt{12994}\\right)$$", "input": "X^{2}=-4-4600\\cdot\\:0.1\\left(X+2\\right)", "steps": [ { "type": "interim", "title": "Expand $$-4-4600\\cdot\\:0.1\\left(X+2\\right):{\\quad}-460X-924$$", "input": "-4-4600\\cdot\\:0.1\\left(X+2\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4600\\cdot\\:0.1=460$$", "result": "=-4-460\\left(X+2\\right)" }, { "type": "interim", "title": "Expand $$-460\\left(X+2\\right):{\\quad}-460X-920$$", "input": "-460\\left(X+2\\right)", "result": "=-4-460X-920", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-460,\\:b=X,\\:c=2$$" ], "result": "=-460X+\\left(-460\\right)\\cdot\\:2", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-460X-460\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$460\\cdot\\:2=920$$", "result": "=-460X-920" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eIdyUl2F0toM2cxX8X9fYt13jtrSFDx+UNsawjlOjV3mCERtN5am/xjLBnVS+YNP8D4yaPBYvrqNvcxJbQLVhPxtgDRvpXZFJM5ns6z004MDaxgiqQQeHO0J6JwCVUFa" } }, { "type": "interim", "title": "Simplify $$-4-460X-920:{\\quad}-460X-924$$", "input": "-4-460X-920", "result": "=-460X-924", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-460X-4-920" }, { "type": "step", "primary": "Subtract the numbers: $$-4-920=-924$$", "result": "=-460X-924" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73itRducAK3A8g7QYwtG2pV3TQj2wDtgWPl2yM5TcGD8LAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2hBHNU81ZxXbnX7i+/GIgBmEUxTkn1jw+5SsnqlaFjRP4=" } }, { "type": "step", "result": "X^{2}=-460X-924" }, { "type": "interim", "title": "Move $$924\\:$$to the left side", "input": "X^{2}=-460X-924", "result": "X^{2}+924=-460X", "steps": [ { "type": "step", "primary": "Add $$924$$ to both sides", "result": "X^{2}+924=-460X-924+924" }, { "type": "step", "primary": "Simplify", "result": "X^{2}+924=-460X" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vlWwbBTRdSFoh4q9Ey3qeRbzM40Tk5qtM06EZ4YoXwejoBEjjQYpCzqL6OG8rnk87q33wQl9f/TJf1Ki5mE9eKJmlSrZ/3f5s6dVhaC2CrUpxg6agBMq+ARA6JD0jh/uMxXBhXC6QwtNJrk64ZxxN9j/RtTYJkwHRMpnvM8A6/UKXES3qYzjDMwmoojjBOzB1U6tmrtv1SqcHAtgI3wbI2TSV74sOmTYBIfKAShLBW4MIv8EMly2RoZ9ZS8QZFS1zLLwgw4W5PQ0CRhtOa+q+NwjO374w6AFeqR9BqI0eauyOi9aTjMQXCZ+0ZF/GeM6X76w3Gg0GKTBnDcDlGpLcGRLd2VwIqlBNByF6663sySTAsNGa+CBJ/KzhLR7JS0jDnXl5zSkEKKRK3f3pyxtGrlDAiBKp3ou2VFdtkXNT6o=" } }, { "type": "interim", "title": "Move $$460X\\:$$to the left side", "input": "X^{2}+924=-460X", "result": "X^{2}+924+460X=0", "steps": [ { "type": "step", "primary": "Add $$460X$$ to both sides", "result": "X^{2}+924+460X=-460X+460X" }, { "type": "step", "primary": "Simplify", "result": "X^{2}+924+460X=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": "X^{2}+460X+924=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "X^{2}+460X+924=0", "result": "{X}_{1,\\:2}=\\frac{-460\\pm\\:\\sqrt{460^{2}-4\\cdot\\:1\\cdot\\:924}}{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 $${\\quad}a=1,\\:b=460,\\:c=924$$", "result": "{X}_{1,\\:2}=\\frac{-460\\pm\\:\\sqrt{460^{2}-4\\cdot\\:1\\cdot\\:924}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{460^{2}-4\\cdot\\:1\\cdot\\:924}=4\\sqrt{12994}$$", "input": "\\sqrt{460^{2}-4\\cdot\\:1\\cdot\\:924}", "result": "{X}_{1,\\:2}=\\frac{-460\\pm\\:4\\sqrt{12994}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:924=3696$$", "result": "=\\sqrt{460^{2}-3696}" }, { "type": "step", "primary": "$$460^{2}=211600$$", "result": "=\\sqrt{211600-3696}" }, { "type": "step", "primary": "Subtract the numbers: $$211600-3696=207904$$", "result": "=\\sqrt{207904}" }, { "type": "interim", "title": "Prime factorization of $$207904:{\\quad}2^{5}\\cdot\\:73\\cdot\\:89$$", "input": "207904", "result": "=\\sqrt{2^{5}\\cdot\\:73\\cdot\\:89}", "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFzN2BAVhx3JVGOlTUgE2k20mV4KYS2geCjKF2tBieK5yC0qhtadDQDf8S0mAbjwz2w97H6gmk50Y1XE61UYXgvY" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{4}\\cdot\\:2\\cdot\\:73\\cdot\\:89}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2^{4}}\\sqrt{2\\cdot\\:73\\cdot\\:89}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^m}=a^{\\frac{m}{n}}$$", "secondary": [ "$$\\sqrt{2^{4}}=2^{\\frac{4}{2}}=2^{2}$$" ], "result": "=2^{2}\\sqrt{2\\cdot\\:73\\cdot\\:89}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=4\\sqrt{12994}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+AAqVWCbQczo1wIkAuUEIZGf1KxrLW44chFgPJhAsD59PD6ky+0+TOWGvQOHH/hmcJChiVhDxT5N/LHSTLMjyEyaBrprlI0J2ZUsJkByZi0/y9DKGIPglJ+qMi9xDu2KGm3HfK+GmWkH7ffUNJxI3xeTh/J9kO8Uxr3GNnQzErppKhxYnaM2m89HiOD+MKJMt3mQM/uY+GtueDZzlXgWOQ==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{X}_{1}=\\frac{-460+4\\sqrt{12994}}{2\\cdot\\:1},\\:{X}_{2}=\\frac{-460-4\\sqrt{12994}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$X=\\frac{-460+4\\sqrt{12994}}{2\\cdot\\:1}:{\\quad}2\\left(\\sqrt{12994}-115\\right)$$", "input": "\\frac{-460+4\\sqrt{12994}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-460+4\\sqrt{12994}}{2}" }, { "type": "interim", "title": "Factor $$-460+4\\sqrt{12994}:{\\quad}4\\left(-115+\\sqrt{12994}\\right)$$", "input": "-460+4\\sqrt{12994}", "result": "=\\frac{4\\left(-115+\\sqrt{12994}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-4\\cdot\\:115+4\\sqrt{12994}" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(-115+\\sqrt{12994}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2\\left(\\sqrt{12994}-115\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yZWZf8rG5JfAmruTbOR/yNOxlOrGSwpiNcoDiFSao8HL/Yp19tRThSsSWFQl2zjFdYPfXQvX4/bINBB8wSEQ0aDkRai/mnDnLMZEbb1o9sLvMELlLPEzLKA+XFJKGqxj8mXEIbHMTQ2XBg4B+Ptp/IbJtTUOD8nT7ByVBadyk63o3lgOgsdAdQYuvodANo9wialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$X=\\frac{-460-4\\sqrt{12994}}{2\\cdot\\:1}:{\\quad}-2\\left(115+\\sqrt{12994}\\right)$$", "input": "\\frac{-460-4\\sqrt{12994}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-460-4\\sqrt{12994}}{2}" }, { "type": "interim", "title": "Factor $$-460-4\\sqrt{12994}:{\\quad}-4\\left(115+\\sqrt{12994}\\right)$$", "input": "-460-4\\sqrt{12994}", "result": "=-\\frac{4\\left(115+\\sqrt{12994}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-4\\cdot\\:115-4\\sqrt{12994}" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=-4\\left(115+\\sqrt{12994}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2\\left(115+\\sqrt{12994}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7c1+7Jk94O17LxGGo9NSxr9OxlOrGSwpiNcoDiFSao8HL/Yp19tRThSsSWFQl2zjFdYPfXQvX4/bINBB8wSEQ0VBpzwGpVFkd2WYPgTJ2KwgtXtN79NWR4EcFiZD9mTjIbyoL3N+B0VpPzVtQKHlpL4oBJCKxw2MnBGIVxKQoa4mRgJvDldZHbwArZad+RsTvbGUhD9+gDQg079F600Xu5Q==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "X=2\\left(\\sqrt{12994}-115\\right),\\:X=-2\\left(115+\\sqrt{12994}\\right)" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "X", "plotRequest": "X^{2}+4+4600\\cdot 0.1\\cdot (X+2)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }