factor 9a^2+8a-56

{ "query": { "display": "factor $$9a^{2}+8a-56$$", "symbolab_question": "FACTOR#factor 9a^{2}+8a-56" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Factor", "default": "9(a-\\frac{2(\\sqrt{130}-2)}{9})(a+\\frac{2(2+\\sqrt{130})}{9})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$9a^{2}+8a-56:{\\quad}9\\left(a-\\frac{2\\left(\\sqrt{130}-2\\right)}{9}\\right)\\left(a+\\frac{2\\left(2+\\sqrt{130}\\right)}{9}\\right)$$", "input": "9a^{2}+8a-56", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=9\\left(a^{2}+\\frac{8a}{9}-\\frac{56}{9}\\right)" }, { "type": "interim", "title": "Factor $$a^{2}+\\frac{8a}{9}-\\frac{56}{9}:{\\quad}\\left(a-\\frac{2\\left(\\sqrt{130}-2\\right)}{9}\\right)\\left(a+\\frac{2\\left(2+\\sqrt{130}\\right)}{9}\\right)$$", "result": "9\\left(a-\\frac{2\\left(\\sqrt{130}-2\\right)}{9}\\right)\\left(a+\\frac{2\\left(2+\\sqrt{130}\\right)}{9}\\right)", "steps": [ { "type": "interim", "title": "Solve $$a^{2}+\\frac{8a}{9}-\\frac{56}{9}=0:{\\quad}a=\\frac{2\\left(\\sqrt{130}-2\\right)}{9},\\:a=-\\frac{2\\left(2+\\sqrt{130}\\right)}{9}$$", "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 $${\\quad}a=1,\\:b=\\frac{8}{9},\\:c=-\\frac{56}{9}{:\\quad}a_{1,\\:2}=\\frac{-\\frac{8}{9}\\pm\\:\\sqrt{\\left(\\frac{8}{9}\\right)^{2}-4\\cdot\\:1\\left(-\\frac{56}{9}\\right)}}{2\\cdot\\:1}$$" }, { "type": "step", "result": "a=\\frac{-\\frac{8}{9}+\\sqrt{\\left(\\frac{8}{9}\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-\\frac{56}{9}\\right)}}{2\\cdot\\:1}=\\frac{2\\left(\\sqrt{130}-2\\right)}{9}" }, { "type": "step", "result": "a=\\frac{-\\frac{8}{9}-\\sqrt{\\left(\\frac{8}{9}\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-\\frac{56}{9}\\right)}}{2\\cdot\\:1}=-\\frac{2\\left(2+\\sqrt{130}\\right)}{9}" }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "a=\\frac{2\\left(\\sqrt{130}-2\\right)}{9},\\:a=-\\frac{2\\left(2+\\sqrt{130}\\right)}{9}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "Polynom can therefore be factored into", "result": "\\left(a-\\left(\\frac{2\\left(\\sqrt{130}-2\\right)}{9}\\right)\\right)\\left(a-\\left(-\\frac{2\\left(2+\\sqrt{130}\\right)}{9}\\right)\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(a-\\frac{2\\left(\\sqrt{130}-2\\right)}{9}\\right)\\left(a+\\frac{2\\left(2+\\sqrt{130}\\right)}{9}\\right)" } ], "meta": { "interimType": "Generic Factor Specific 1Eq" } } ], "meta": { "solvingClass": "Solver" } }, "meta": { "showVerify": true } }