x^2-14x+36=0

{ "query": { "display": "$$x^{2}-14x+36=0$$", "symbolab_question": "EQUATION#x^{2}-14x+36=0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "x=7+\\sqrt{13},x=7-\\sqrt{13}", "decimal": "x=10.60555…,x=3.39444…", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$x^{2}-14x+36=0$$", "symbolab_question": "quadraticformula x^{2}-14x+36=0" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$x^{2}-14x+36=0$$", "symbolab_question": "completesquare x^{2}-14x+36=0" } } ], "steps": { "type": "interim", "title": "$$x^{2}-14x+36=0{\\quad:\\quad}x=7+\\sqrt{13},\\:x=7-\\sqrt{13}$$", "input": "x^{2}-14x+36=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}-14x+36=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-14\\right)\\pm\\:\\sqrt{\\left(-14\\right)^{2}-4\\cdot\\:1\\cdot\\:36}}{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 $${\\quad}a=1,\\:b=-14,\\:c=36$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-14\\right)\\pm\\:\\sqrt{\\left(-14\\right)^{2}-4\\cdot\\:1\\cdot\\:36}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-14\\right)^{2}-4\\cdot\\:1\\cdot\\:36}=2\\sqrt{13}$$", "input": "\\sqrt{\\left(-14\\right)^{2}-4\\cdot\\:1\\cdot\\:36}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-14\\right)\\pm\\:2\\sqrt{13}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-14\\right)^{2}=14^{2}$$" ], "result": "=\\sqrt{14^{2}-4\\cdot\\:1\\cdot\\:36}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:36=144$$", "result": "=\\sqrt{14^{2}-144}" }, { "type": "step", "primary": "$$14^{2}=196$$", "result": "=\\sqrt{196-144}" }, { "type": "step", "primary": "Subtract the numbers: $$196-144=52$$", "result": "=\\sqrt{52}" }, { "type": "interim", "title": "Prime factorization of $$52:{\\quad}2^{2}\\cdot\\:13$$", "input": "52", "result": "=\\sqrt{2^{2}\\cdot\\:13}", "steps": [ { "type": "step", "primary": "$$52\\:$$divides by $$2\\quad\\:52=26\\cdot\\:2$$", "result": "=2\\cdot\\:26" }, { "type": "step", "primary": "$$26\\:$$divides by $$2\\quad\\:26=13\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:13" }, { "type": "step", "primary": "$$2,\\:13$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:13" }, { "type": "step", "result": "=2^{2}\\cdot\\:13" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRjl/dE9e0owjU0NK6lxSAv7mDmiXg+V79OVxDmBC/OzVB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEAAdoTSIf9EhWkrtoMUM8D/" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{13}\\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{13}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ETEBU5RYU5U/sGxJEz+8lEFj7CiO8FQNDSt2Nz1mnF749mbjgUxe4+sgPetom8YQq47vuWedXv2WUg94ER8IwUj7eM16HL8BLy8LPr9glcWjeh7+jKEzLb7VNCEMF3Z/wyyq06JFQE1pxa75631ctcr4i+SWxENyYmpjB8Gs1ej0jI5Bhnn+SuaFUUKfmo7hJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-14\\right)+2\\sqrt{13}}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-\\left(-14\\right)-2\\sqrt{13}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-14\\right)+2\\sqrt{13}}{2\\cdot\\:1}:{\\quad}7+\\sqrt{13}$$", "input": "\\frac{-\\left(-14\\right)+2\\sqrt{13}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{14+2\\sqrt{13}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{14+2\\sqrt{13}}{2}" }, { "type": "interim", "title": "Factor $$14+2\\sqrt{13}:{\\quad}2\\left(7+\\sqrt{13}\\right)$$", "input": "14+2\\sqrt{13}", "result": "=\\frac{2\\left(7+\\sqrt{13}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:7+2\\sqrt{13}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(7+\\sqrt{13}\\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": "=7+\\sqrt{13}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/p8HuK7VGPkknYCaz2jCFYqiJVdZMJVBqbC9Uv4iz7xoHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJTMBCQoSBOeOpWhIlf4nUC0/y9DKGIPglJ+qMi9xDu2KDVgJU+IgTXMRRpCGlExaQYqiJVdZMJVBqbC9Uv4iz7x3sNoh6cotOLBm/ZqdLDCN" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-14\\right)-2\\sqrt{13}}{2\\cdot\\:1}:{\\quad}7-\\sqrt{13}$$", "input": "\\frac{-\\left(-14\\right)-2\\sqrt{13}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{14-2\\sqrt{13}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{14-2\\sqrt{13}}{2}" }, { "type": "interim", "title": "Factor $$14-2\\sqrt{13}:{\\quad}2\\left(7-\\sqrt{13}\\right)$$", "input": "14-2\\sqrt{13}", "result": "=\\frac{2\\left(7-\\sqrt{13}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:7-2\\sqrt{13}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(7-\\sqrt{13}\\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": "=7-\\sqrt{13}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fVA7qbRcSLybWuKD7nrUiIqiJVdZMJVBqbC9Uv4iz7xoHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJXqZsYmKDcby/n12bLL60JU/y9DKGIPglJ+qMi9xDu2KmEA51BcxXY/5UoLW1mgKVoqiJVdZMJVBqbC9Uv4iz7x3sNoh6cotOLBm/ZqdLDCN" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=7+\\sqrt{13},\\:x=7-\\sqrt{13}" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}-14x+36" }, "showViewLarger": true } }, "meta": { "showVerify": true } }