x^2-2x=0

{ "query": { "display": "$$x^{2}-2x=0$$", "symbolab_question": "EQUATION#x^{2}-2x=0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Quadratic", "default": "x=2,x=0", "meta": { "showVerify": true } }, "methods": [ { "method": "Solve with the quadratic formula", "query": { "display": "quadratic formula $$x^{2}-2x=0$$", "symbolab_question": "quadraticformula x^{2}-2x=0" } }, { "method": "Solve by completing the square", "query": { "display": "complete the square $$x^{2}-2x=0$$", "symbolab_question": "completesquare x^{2}-2x=0" } }, { "method": "Solve by factoring", "query": { "display": "factor $$x^{2}-2x=0$$", "symbolab_question": "factor x^{2}-2x=0" } } ], "steps": { "type": "interim", "title": "$$x^{2}-2x=0{\\quad:\\quad}x=2,\\:x=0$$", "input": "x^{2}-2x=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}-2x=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:0}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=-2,\\:c=0$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:0}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:0}=2$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:0}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:2}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{2^{2}-0}" }, { "type": "step", "primary": "$$2^{2}-0=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTnGW/jTwUrqHAcqr1cHDsBCGcE76jDOroaIH3gr3HS6TzMFYmi1F5Hg/ibpEToVnYwSc/te/iOpB2yBHqsIcuJnPYvWlDOvQwlh4TEmetdSScZb+NPBSuocByqvVwcOwEFXNpVIlvpgHOi/3rrWD6As=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-2\\right)+2}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-\\left(-2\\right)-2}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-2\\right)+2}{2\\cdot\\:1}:{\\quad}2$$", "input": "\\frac{-\\left(-2\\right)+2}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\frac{4}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fv9gVNWyI/X8TvzzSZV8n7JqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUDMAhHkxmnK9YrvcjcFZXqVhVjXCvkuV0dlNfJSARrUY1FYlXu7xkLLsgM93YHqktw==" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-2\\right)-2}{2\\cdot\\:1}:{\\quad}0$$", "input": "\\frac{-\\left(-2\\right)-2}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$2-2=0$$", "result": "=\\frac{0}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzZ+Rg5UPtRAM2q6UZuzBrJqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09jMAhHkxmnK9YrvcjcFZXqUEhkNIoEcCE5KusvGzEjW21FYlXu7xkLLsgM93YHqktw==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=2,\\:x=0" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/quadratic-equations-practice", "practiceTopic": "Quadratic Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}-2x" }, "showViewLarger": true } }, "meta": { "showVerify": true } }