x^2+10x-y+15=0

{ "query": { "display": "$$x^{2}+10x-y+15=0$$", "symbolab_question": "CONIC#x^{2}+10x-y+15=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-5,-10),p=\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$x^{2}+10x-y+15=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-5,\\:-10\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "x^{2}+10x-y+15=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 $$x^{2}+10x-y+15=0\\:$$in the standard form", "input": "x^{2}+10x-y+15=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "y=x^{2}+10x+15" }, { "type": "interim", "title": "Complete the square $$x^{2}+10x+15:{\\quad}\\left(x+5\\right)^{2}-10$$", "input": "x^{2}+10x+15", "steps": [ { "type": "step", "primary": "Write $$x^{2}+10x+15\\:$$in the form: $$x^2+2ax+a^2$$" }, { "type": "interim", "title": "$$2a=10{\\quad:\\quad}a=5$$", "input": "2a=10", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=10", "result": "a=5", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{10}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=5" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$5^{2}\\:$$", "result": "x^{2}+10x+15+5^{2}-5^{2}" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}+10x+5^{2}=\\left(x+5\\right)^{2}$$", "Complete the square" ], "result": "\\left(x+5\\right)^{2}+15-5^{2}" }, { "type": "step", "primary": "Simplify", "result": "\\left(x+5\\right)^{2}-10" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=\\left(x+5\\right)^{2}-10" }, { "type": "step", "primary": "Add $$10$$ to both sides", "result": "y+10=\\left(x+5\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\cdot\\:\\frac{1}{4}\\left(y-\\left(-10\\right)\\right)=\\left(x-\\left(-5\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\cdot\\:\\frac{1}{4}\\left(y-\\left(-10\\right)\\right)=\\left(x-\\left(-5\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-5,\\:-10\\right),\\:p=\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-5))^{2}}{4\\frac{1}{4}}-10", "displayFormula": "4\\frac{1}{4}(y-(-10))=(x-(-5))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{41}{4}", "displayFormula": "y=-\\frac{41}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-5,-10)", "(-5,-\\frac{39}{4})" ], "pointsDecimal": [ { "fst": -5, "snd": -10 }, { "fst": -5, "snd": -9.75 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4\\cdot \\frac{1}{4}(y-(-10))=(x-(-5))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -10.428571428571429, "xMax": 5.285714285714285, "yMin": -12.571428571428571, "yMax": 3.1428571428571423 } }, "showViewLarger": true } } }