18x^2-64x-14y+150=0

{ "query": { "display": "$$18x^{2}-64x-14y+150=0$$", "symbolab_question": "CONIC#18x^{2}-64x-14y+150=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(\\frac{16}{9},\\frac{419}{63}),p=\\frac{7}{36}" }, "steps": { "type": "interim", "title": "$$18x^{2}-64x-14y+150=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(\\frac{16}{9},\\:\\frac{419}{63}\\right),\\:$$and focal length $$|p|=\\frac{7}{36}$$", "input": "18x^{2}-64x-14y+150=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 $$18x^{2}-64x-14y+150=0\\:$$in the standard form", "input": "18x^{2}-64x-14y+150=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "14y=18x^{2}-64x+150" }, { "type": "step", "primary": "Divide by $$14$$", "result": "y=\\frac{9x^{2}}{7}-\\frac{32x}{7}+\\frac{75}{7}" }, { "type": "interim", "title": "Complete the square $$\\frac{9x^{2}}{7}-\\frac{32x}{7}+\\frac{75}{7}:{\\quad}\\frac{9}{7}\\left(x-\\frac{16}{9}\\right)^{2}+\\frac{419}{63}$$", "input": "\\frac{9x^{2}}{7}-\\frac{32x}{7}+\\frac{75}{7}", "steps": [ { "type": "step", "primary": "Write $$\\frac{9x^{2}}{7}-\\frac{32x}{7}+\\frac{75}{7}\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$\\frac{9}{7}$$" ], "result": "\\frac{9}{7}\\left(x^{2}-\\frac{32x}{9}+\\frac{25}{3}\\right)" }, { "type": "interim", "title": "$$2a=-\\frac{32}{9}{\\quad:\\quad}a=-\\frac{16}{9}$$", "input": "2a=-\\frac{32}{9}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-\\frac{32}{9}", "result": "a=-\\frac{16}{9}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-\\frac{32}{9}}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{-\\frac{32}{9}}{2}", "result": "a=-\\frac{16}{9}", "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 $$\\frac{-\\frac{32}{9}}{2}:{\\quad}-\\frac{16}{9}$$", "input": "\\frac{-\\frac{32}{9}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{32}{9}}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{32}{9}}{2}=\\frac{32}{9\\cdot\\:2}$$" ], "result": "=-\\frac{32}{9\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:2=18$$", "result": "=-\\frac{32}{18}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{16}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVYnkKe8obuq738IrYnsRyrDdd47a0hQ8flDbGsI5To1dsabmmR9ZBQ+yLvQJaWqH0PsrRBCls8pEfZdTWHr3xVseNvb7k0sVmuwf19w9aD9NwFkJl/hQGUOe3kPB8GwiQu2NyC3uZMvKSSsjgNHC4yGPATzsdMJbIBTfKlGugMwN" } }, { "type": "step", "result": "a=-\\frac{16}{9}" } ], "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 $$\\left(-\\frac{16}{9}\\right)^{2}\\:$$", "result": "\\frac{9}{7}\\left(x^{2}-\\frac{32x}{9}+\\frac{25}{3}+\\left(-\\frac{16}{9}\\right)^{2}-\\left(-\\frac{16}{9}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-\\frac{32}{9}x+\\left(-\\frac{16}{9}\\right)^{2}=\\left(x-\\frac{16}{9}\\right)^{2}$$", "Complete the square" ], "result": "\\frac{9}{7}\\left(\\left(x-\\frac{16}{9}\\right)^{2}+\\frac{25}{3}-\\left(-\\frac{16}{9}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{9}{7}\\left(x-\\frac{16}{9}\\right)^{2}+\\frac{419}{63}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=\\frac{9}{7}\\left(x-\\frac{16}{9}\\right)^{2}+\\frac{419}{63}" }, { "type": "step", "primary": "Subtract $$\\frac{419}{63}$$ from both sides", "result": "y-\\frac{419}{63}=\\frac{9}{7}\\left(x-\\frac{16}{9}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$\\frac{9}{7}$$", "result": "\\frac{7}{9}\\left(y-\\frac{419}{63}\\right)=\\left(x-\\frac{16}{9}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\cdot\\:\\frac{7}{36}\\left(y-\\frac{419}{63}\\right)=\\left(x-\\frac{16}{9}\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\cdot\\:\\frac{7}{36}\\left(y-\\frac{419}{63}\\right)=\\left(x-\\frac{16}{9}\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(\\frac{16}{9},\\:\\frac{419}{63}\\right),\\:p=\\frac{7}{36}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-\\frac{16}{9})^{2}}{4\\frac{7}{36}}+\\frac{419}{63}", "displayFormula": "4\\frac{7}{36}(y-\\frac{419}{63})=(x-\\frac{16}{9})^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{1627}{252}", "displayFormula": "y=\\frac{1627}{252}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" 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