6x^2-27=0

{ "query": { "display": "$$6x^{2}-27=0$$", "symbolab_question": "EQUATION#6x^{2}-27=0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "x=\\frac{3}{\\sqrt{2}},x=-\\frac{3}{\\sqrt{2}}", "decimal": "x=2.12132…,x=-2.12132…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$6x^{2}-27=0{\\quad:\\quad}x=\\frac{3}{\\sqrt{2}},\\:x=-\\frac{3}{\\sqrt{2}}$$", "input": "6x^{2}-27=0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "6x^{2}-27=0", "result": "6x^{2}=27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "6x^{2}-27+27=0+27" }, { "type": "step", "primary": "Simplify", "result": "6x^{2}=27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x^{2}=27", "result": "x^{2}=\\frac{9}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x^{2}}{6}=\\frac{27}{6}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=\\frac{9}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "x=\\sqrt{\\frac{9}{2}},\\:x=-\\sqrt{\\frac{9}{2}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{9}{2}}=\\frac{3}{\\sqrt{2}}$$", "input": "\\sqrt{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{9}}{\\sqrt{2}}" }, { "type": "interim", "title": "$$\\sqrt{9}=3$$", "input": "\\sqrt{9}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74zYoh2RHt5bF5LAil06rQ913jtrSFDx+UNsawjlOjV0Fuo8niSX4SUBqNT4QFNxbcubCnYZOJ5L8/2gsdymw1KIDQ0pFG0qBgUJZZplUxb0=" } }, { "type": "step", "result": "=\\frac{3}{\\sqrt{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0NwP7If4Z3q+v8+wwJj4STsSrju+5Z51e/ZZSD3gRHwjBbCldzqCzpejBNnXFLKb9rhTvQPornS61GDWTUYpUdOOKy+4oc4WkS69iOkw+tuw7U2YDtx8mJZlPbqXzONnYf9otHBlzhTAO7o5XztqTTkGwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{9}{2}}=-\\frac{3}{\\sqrt{2}}$$", "input": "-\\sqrt{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\:\\quad\\:a\\ge0,\\:b\\ge0$$", "result": "=-\\frac{\\sqrt{9}}{\\sqrt{2}}" }, { "type": "interim", "title": "$$\\sqrt{9}=3$$", "input": "\\sqrt{9}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74zYoh2RHt5bF5LAil06rQ913jtrSFDx+UNsawjlOjV0Fuo8niSX4SUBqNT4QFNxbcubCnYZOJ5L8/2gsdymw1KIDQ0pFG0qBgUJZZplUxb0=" } }, { "type": "step", "result": "=-\\frac{3}{\\sqrt{2}}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+Arxm5zuS/84yhgYFbWqXxKY3QIJQJZuTAY5js+oqjdT8ksl5/9zM83eMybuFrx+1vgha3bKyzRGn37jQ8thU9HS1iRrXh3OSOXhogkv14GjeoU64tJRthB+J/MF+wJHnPsExj6f5BxtPnmY1A9eRK+s1dFYlSN7rzdW8vgDfTX46ChV" } }, { "type": "step", "result": "x=\\frac{3}{\\sqrt{2}},\\:x=-\\frac{3}{\\sqrt{2}}" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "6x^{2}-27" }, "showViewLarger": true } }, "meta": { "showVerify": true } }