2x^2-4x+y-3=0

{ "query": { "display": "$$2x^{2}-4x+y-3=0$$", "symbolab_question": "CONIC#2x^{2}-4x+y-3=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(1,5),p=-\\frac{1}{8}" }, "steps": { "type": "interim", "title": "$$2x^{2}-4x+y-3=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(1,\\:5\\right),\\:$$and focal length $$|p|=\\frac{1}{8}$$", "input": "2x^{2}-4x+y-3=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 $$2x^{2}-4x+y-3=0\\:$$in the standard form", "input": "2x^{2}-4x+y-3=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "y=-2x^{2}+4x+3" }, { "type": "interim", "title": "Complete the square $$-2x^{2}+4x+3:{\\quad}-2\\left(x-1\\right)^{2}+5$$", "input": "-2x^{2}+4x+3", "steps": [ { "type": "step", "primary": "Write $$-2x^{2}+4x+3\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-2$$" ], "result": "-2\\left(x^{2}-2x-\\frac{3}{2}\\right)" }, { "type": "interim", "title": "$$2a=-2{\\quad:\\quad}a=-1$$", "input": "2a=-2", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-2", "result": "a=-1", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-2}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=-1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-1\\right)^{2}\\:$$", "result": "-2\\left(x^{2}-2x-\\frac{3}{2}+\\left(-1\\right)^{2}-\\left(-1\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-2x+\\left(-1\\right)^{2}=\\left(x-1\\right)^{2}$$", "Complete the square" ], "result": "-2\\left(\\left(x-1\\right)^{2}-\\frac{3}{2}-\\left(-1\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-2\\left(x-1\\right)^{2}+5" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=-2\\left(x-1\\right)^{2}+5" }, { "type": "step", "primary": "Subtract $$5$$ from both sides", "result": "y-5=-2\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-2$$", "result": "-\\frac{1}{2}\\left(y-5\\right)=\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{8}\\right)\\left(y-5\\right)=\\left(x-1\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{8}\\right)\\left(y-5\\right)=\\left(x-1\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(1,\\:5\\right),\\:p=-\\frac{1}{8}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-1)^{2}}{4(-\\frac{1}{8})}+5", "displayFormula": "4(-\\frac{1}{8})(y-5)=(x-1)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{41}{8}", "displayFormula": "y=\\frac{41}{8}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,5)", "(1,\\frac{39}{8})" ], "pointsDecimal": [ { "fst": 1, "snd": 5 }, { "fst": 1, "snd": 4.875 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-\\frac{1}{8})(y-5)=(x-1)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -3.2857142857142856, "xMax": 4.571428571428571, "yMin": -1.5714285714285712, "yMax": 6.285714285714286 } }, "showViewLarger": true } } }