3x^2+5x-k(x)=0

{ "query": { "display": "$$3x^{2}+5x-k\\left(x\\right)=0$$", "symbolab_question": "CONIC#3x^{2}+5x-k(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{5}{6},-\\frac{25}{12}),p=\\frac{1}{12}" }, "steps": { "type": "interim", "title": "$$3x^{2}+5x-y=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{5}{6},\\:-\\frac{25}{12}\\right),\\:$$and focal length $$|p|=\\frac{1}{12}$$", "input": "3x^{2}+5x-y=0", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ is the standard equation for an up-down facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$3x^{2}+5x-y=0\\:$$in the standard form", "input": "3x^{2}+5x-y=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "y=3x^{2}+5x" }, { "type": "interim", "title": "Complete the square $$3x^{2}+5x:{\\quad}3\\left(x+\\frac{5}{6}\\right)^{2}-\\frac{25}{12}$$", "input": "3x^{2}+5x", "steps": [ { "type": "step", "primary": "Write $$3x^{2}+5x\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$3$$" ], "result": "3\\left(x^{2}+\\frac{5x}{3}\\right)" }, { "type": "interim", "title": "$$2a=\\frac{5}{3}{\\quad:\\quad}a=\\frac{5}{6}$$", "input": "2a=\\frac{5}{3}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=\\frac{5}{3}", "result": "a=\\frac{5}{6}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{\\frac{5}{3}}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{\\frac{5}{3}}{2}", "result": "a=\\frac{5}{6}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2a}{2}:{\\quad}a$$", "input": "\\frac{2a}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n/9TIosDfOyZXH4usT9jsy061ljBSPJeENOw2efoSWuFFDuzzgIzA1F36c+SAFGgRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pQyGeXuoJHAaVrX92ZjOTT" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5}{3}}{2}:{\\quad}\\frac{5}{6}$$", "input": "\\frac{\\frac{5}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\frac{5}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdOhMiTfGswczRF8hIUIlWMDnzlbPZjyKgy1eUCFsLd5YSjT6Of+O++tJijO/Dvck6vKzRUWvR7H7TI+qhAXIp0eKYFFhGodJZ4mS5Q+F9MLRBHnX1v95RIc5nuXPcUCVzQJlT9qiZDPdH9NYNoNRIs=" } }, { "type": "step", "result": "a=\\frac{5}{6}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(\\frac{5}{6}\\right)^{2}\\:$$", "result": "3\\left(x^{2}+\\frac{5x}{3}+\\left(\\frac{5}{6}\\right)^{2}-\\left(\\frac{5}{6}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}+\\frac{5}{3}x+\\left(\\frac{5}{6}\\right)^{2}=\\left(x+\\frac{5}{6}\\right)^{2}$$", "Complete the square" ], "result": "3\\left(\\left(x+\\frac{5}{6}\\right)^{2}-\\left(\\frac{5}{6}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "3\\left(x+\\frac{5}{6}\\right)^{2}-\\frac{25}{12}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=3\\left(x+\\frac{5}{6}\\right)^{2}-\\frac{25}{12}" }, { "type": "step", "primary": "Add $$\\frac{25}{12}$$ to both sides", "result": "y+\\frac{25}{12}=3\\left(x+\\frac{5}{6}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$3$$", "result": "\\frac{1}{3}\\left(y+\\frac{25}{12}\\right)=\\left(x+\\frac{5}{6}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\cdot\\:\\frac{1}{12}\\left(y-\\left(-\\frac{25}{12}\\right)\\right)=\\left(x-\\left(-\\frac{5}{6}\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\cdot\\:\\frac{1}{12}\\left(y-\\left(-\\frac{25}{12}\\right)\\right)=\\left(x-\\left(-\\frac{5}{6}\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{5}{6},\\:-\\frac{25}{12}\\right),\\:p=\\frac{1}{12}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-\\frac{5}{6}))^{2}}{4\\frac{1}{12}}-\\frac{25}{12}", "displayFormula": 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