|3x^2+16|=2

{ "query": { "display": "$$\\left|3x^{2}+16\\right|=2$$", "symbolab_question": "EQUATION#\\left|3x^{2}+16\\right|=2" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Absolute", "default": "\\mathrm{No\\:Solution}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left|3x^{2}+16\\right|=2{\\quad:\\quad}$$No Solution", "input": "\\left|3x^{2}+16\\right|=2", "steps": [ { "type": "step", "primary": "Apply absolute rule: If $$|u|\\:=\\:a,\\:a>0\\:$$then $$u\\:=\\:a\\:\\quad\\:$$or$$\\quad\\:u\\:=\\:-a$$", "result": "3x^{2}+16=-2\\lor\\:3x^{2}+16=2" }, { "type": "interim", "title": "$$3x^{2}+16=-2\\lor\\:3x^{2}+16=2$$", "input": "3x^{2}+16=-2\\lor\\:3x^{2}+16=2", "result": "\\mathrm{No\\:Solution}\\lor\\:\\mathrm{No\\:Solution}", "steps": [ { "type": "interim", "title": "$$3x^{2}+16=-2{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "3x^{2}+16=-2", "steps": [ { "type": "interim", "title": "Move $$16\\:$$to the right side", "input": "3x^{2}+16=-2", "result": "3x^{2}=-18", "steps": [ { "type": "step", "primary": "Subtract $$16$$ from both sides", "result": "3x^{2}+16-16=-2-16" }, { "type": "step", "primary": "Simplify", "result": "3x^{2}=-18" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x^{2}=-18", "result": "x^{2}=-6", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x^{2}}{3}=\\frac{-18}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x^{2}}{3}=\\frac{-18}{3}", "result": "x^{2}=-6", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x^{2}}{3}:{\\quad}x^{2}$$", "input": "\\frac{3x^{2}}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SpwciAZvlOxJGLnGBB5g4RZsnww+P7XjdokU4IcmGg+jkVi15I8rBefLi4Iyt2wrDzRG5i2RnXJK7EZvbhYwA4saY2cs9YZ/08feWmaOgkgpmw6HumtOtBdZArFlLt0QcEuadShQm8Wq4nubtzCpdg==" } }, { "type": "interim", "title": "Simplify $$\\frac{-18}{3}:{\\quad}-6$$", "input": "\\frac{-18}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{18}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{18}{3}=6$$", "result": "=-6" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77NU3bzHZfl3cZQRcH5pkpACWKUbvV6WK3fDUgFtg3Q8IVlRpyxl4O+CcU61LkQcQZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz24FIzciTPvX/24X5LhWxo/sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x^{2}=-6" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$3x^{2}+16=2{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "3x^{2}+16=2", "steps": [ { "type": "interim", "title": "Move $$16\\:$$to the right side", "input": "3x^{2}+16=2", "result": "3x^{2}=-14", "steps": [ { "type": "step", "primary": "Subtract $$16$$ from both sides", "result": "3x^{2}+16-16=2-16" }, { "type": "step", "primary": "Simplify", "result": "3x^{2}=-14" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x^{2}=-14", "result": "x^{2}=-\\frac{14}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x^{2}}{3}=\\frac{-14}{3}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-\\frac{14}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Combine Solutions:", "result": "\\mathrm{No\\:Solution}\\lor\\:\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\left|3x^{2}+16\\right|-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }