intercepts of xsqrt(36-x^2)

{ "query": { "display": "intercepts $$x\\sqrt{36-x^{2}}$$", "symbolab_question": "CONIC#intercepts x\\sqrt{36-x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{X\\:Intercepts}: (0,0),(6,0),(-6,0),\\mathrm{Y\\:Intercepts}: (0,0)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$x\\sqrt{36-x^{2}}:\\quad\\:$$X Intercepts$$:\\:\\left(0,\\:0\\right),\\:\\left(6,\\:0\\right),\\:\\left(-6,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$x\\sqrt{36-x^{2}}:{\\quad}\\left(0,\\:0\\right),\\:\\left(6,\\:0\\right),\\:\\left(-6,\\:0\\right)$$", "input": "x\\sqrt{36-x^{2}}", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$x\\sqrt{36-x^{2}}=0:{\\quad}x=0,\\:x=6,\\:x=-6$$", "input": "x\\sqrt{36-x^{2}}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x=0\\lor\\:\\sqrt{36-x^{2}}=0" }, { "type": "interim", "title": "Solve $$\\sqrt{36-x^{2}}=0:{\\quad}x=6,\\:x=-6$$", "input": "\\sqrt{36-x^{2}}=0", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}36-x^{2}=0$$", "input": "\\sqrt{36-x^{2}}=0", "result": "36-x^{2}=0", "steps": [ { "type": "step", "result": "\\left(\\sqrt{36-x^{2}}\\right)^{2}=0^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{36-x^{2}}\\right)^{2}:{\\quad}36-x^{2}$$", "input": "\\left(\\sqrt{36-x^{2}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(36-x^{2}\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(36-x^{2}\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=36-x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAifbUYmNH9wXtF4IykdnkM8cLAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2hxFuPbip5EmRsABmWzH0mIGDvfRP7tC7wJYzgNs+0Xl4=" } }, { "type": "interim", "title": "Expand $$0^{2}:{\\quad}0$$", "input": "0^{2}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y+JsXs5Wey1G4iayV5SJuA7afGBEifydK7wCwslogi9F5bLdlaCFshD3v4rLy7EcFU0qZ4XMwhSlm3knFkjDUN7UD16RqFQha281as3i/3hi3DJlF4JtuHZL1PI6pH9O" } }, { "type": "step", "result": "36-x^{2}=0" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWTbwNOjrvAfIM2wCGzb8QpuuQTC2rJk2wnzJ36NyzAtVY86dFSmxhIBSkspfk/4gY2pwARdbpc94GT6N/QUat9Hql8XXPq6bNQlMm+36iNhmgLTdIEuzwdV+ksVN+7oK2hn7L5gTBUzF4g+U/sLR6Sg==" } }, { "type": "interim", "title": "Solve $$36-x^{2}=0:{\\quad}x=6,\\:x=-6$$", "input": "36-x^{2}=0", "steps": [ { "type": "interim", "title": "Move $$36\\:$$to the right side", "input": "36-x^{2}=0", "result": "-x^{2}=-36", "steps": [ { "type": "step", "primary": "Subtract $$36$$ from both sides", "result": "36-x^{2}-36=0-36" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}=-36" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-x^{2}=-36", "result": "x^{2}=36", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-x^{2}}{-1}=\\frac{-36}{-1}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=36" } ], "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{36},\\:x=-\\sqrt{36}" }, { "type": "interim", "title": "$$\\sqrt{36}=6$$", "input": "\\sqrt{36}", "steps": [ { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ba5g+UZ5roMy68ii/TX6lSAn9lkDfZkicUGkO3EF+IowGPWQQxj+Buy70vvMPVbRzGQBGgzjhY9wy62Ypaxu7DP9zNOAin/+9QrppOujlp4=" } }, { "type": "interim", "title": "$$-\\sqrt{36}=-6$$", "input": "-\\sqrt{36}", "steps": [ { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=-\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{6^{2}}=-6$$" ], "result": "=-6" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qIDYfCJvFLiSiwojgehIwN6GQqufR6tr2vPxOUv7H++DRmTmvBPRvFp0k+uWRDqurAJFSr9Yxv/LaHr9YfwzxnHdXbwYdlu4HlGbbwcRG4U=" } }, { "type": "step", "result": "x=6,\\:x=-6" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=6,\\:x=-6" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=6\\:$$True$$,\\:\\:x=-6\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{36-x^{2}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=6:{\\quad}$$True", "input": "\\sqrt{36-6^{2}}=0", "steps": [ { "type": "interim", "title": "$$\\sqrt{36-6^{2}}=0$$", "input": "\\sqrt{36-6^{2}}", "steps": [ { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-36}" }, { "type": "step", "primary": "Subtract the numbers: $$36-36=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75F9w8AeG+mABRa0zCFuUD0V2IovRuj7SUtFTM1ebUgQE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7tisCJ6DKljm4cpqqTTXla/rkQRYlc0DxGPn21AQgIWYWg" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } }, { "type": "interim", "title": "Plug in $$x=-6:{\\quad}$$True", "input": "\\sqrt{36-\\left(-6\\right)^{2}}=0", "steps": [ { "type": "interim", "title": "$$\\sqrt{36-\\left(-6\\right)^{2}}=0$$", "input": "\\sqrt{36-\\left(-6\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-6\\right)^{2}=6^{2}$$" ], "result": "=\\sqrt{36-6^{2}}" }, { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-36}" }, { "type": "step", "primary": "Subtract the numbers: $$36-36=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vRvbRh1r/cyY5y8Pn9lbl11iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX4H3s6AQN3slQ8LSXSpgD95VfrHM/WjtlmmD/uTQMUgf" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "x=6,\\:x=-6" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=0,\\:x=6,\\:x=-6" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=0\\:$$True$$,\\:\\:x=6\\:$$True$$,\\:\\:x=-6\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$x\\sqrt{36-x^{2}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=0:{\\quad}$$True", "input": "0\\cdot\\:\\sqrt{36-0^{2}}=0", "steps": [ { "type": "interim", "title": "$$0\\cdot\\:\\sqrt{36-0^{2}}=0$$", "input": "0\\cdot\\:\\sqrt{36-0^{2}}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=0\\cdot\\:\\sqrt{36-0}" }, { "type": "interim", "title": "$$\\sqrt{36-0}=6$$", "input": "\\sqrt{36-0}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$36-0=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aEYi9wNujtaF0xOMmskYdVXTSum/z5kLpMzXS1UJIexyk0R8oZWybxT1ILwFJgqrhj6ylSwHjJkmQPKH34+7f385/YWN2Y6UOI+1UQSUpsQ=" } }, { "type": "step", "result": "=0\\cdot\\:6" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Y1o1ilPOV2x9Tl7301p1GMc9800WW+LP4/8QiFQJd8XNGoPE9TME3q+OPmgkv2RQz7FeFrf3rcM8IZlDz2c0ds/YYA6t7QSfQHToEDpf5ENRf8NNi22Bpta3LRZhwGozEEWJXNA8Rj59tQEICFmFoA==" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } }, { "type": "interim", "title": "Plug in $$x=6:{\\quad}$$True", "input": "6\\sqrt{36-6^{2}}=0", "steps": [ { "type": "interim", "title": "$$6\\sqrt{36-6^{2}}=0$$", "input": "6\\sqrt{36-6^{2}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{36-6^{2}}=0$$", "input": "\\sqrt{36-6^{2}}", "steps": [ { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-36}" }, { "type": "step", "primary": "Subtract the numbers: $$36-36=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75F9w8AeG+mABRa0zCFuUD0V2IovRuj7SUtFTM1ebUgQE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7tisCJ6DKljm4cpqqTTXla/rkQRYlc0DxGPn21AQgIWYWg" } }, { "type": "step", "result": "=6\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mTVHZh5G4vJJXqdlNeRwF1TiRdarS5WaUwjWcvuHZnijkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhC60kSHXclPApRuAjDolukjcheOfdo7P8OQss8x+Zbe2" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } }, { "type": "interim", "title": "Plug in $$x=-6:{\\quad}$$True", "input": "\\left(-6\\right)\\sqrt{36-\\left(-6\\right)^{2}}=0", "steps": [ { "type": "interim", "title": "$$\\left(-6\\right)\\sqrt{36-\\left(-6\\right)^{2}}=0$$", "input": "\\left(-6\\right)\\sqrt{36-\\left(-6\\right)^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-6\\sqrt{36-\\left(-6\\right)^{2}}" }, { "type": "interim", "title": "$$\\sqrt{36-\\left(-6\\right)^{2}}=0$$", "input": "\\sqrt{36-\\left(-6\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(-6\\right)^{2}=36$$", "input": "\\left(-6\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-6\\right)^{2}=6^{2}$$" ], "result": "=6^{2}" }, { "type": "step", "primary": "$$6^{2}=36$$", "result": "=36" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gpGzC1IzTRiHQ4tLcv719s0ag8T1MwTer44+aCS/ZFAwitNRZzazeCL+a30KIiFheA5z3UKDXcbGAxZLpIbQjnONFx3hYQoNAOtg0fmDtn4=" } }, { "type": "step", "result": "=\\sqrt{36-36}" }, { "type": "step", "primary": "Subtract the numbers: $$36-36=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vRvbRh1r/cyY5y8Pn9lbl11iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX4H3s6AQN3slQ8LSXSpgD95VfrHM/WjtlmmD/uTQMUgf" } }, { "type": "step", "result": "=-6\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74VfcTnL0VgZNOOtfum2/avQ57jCj5klqkxZEDQNftKMDnzlbPZjyKgy1eUCFsLd5uA0vbJw0XRhSBCaIjQqkFvvDPxhuAi6BImEOeLcn6xvK9NLdT2JTcjGstJUwbKq+sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "x=0,\\:x=6,\\:x=-6" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right),\\:\\left(6,\\:0\\right),\\:\\left(-6,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7M3Y357aig87h4JRAWSmQBXu3tur4uYXhnv3yafWf9Yvg19JPhDP9L8Df2H8AV9HbuSmyjUT9pVJ6Sq3lZi1y4MTE/1Yf/XTpqQ1Qnvjddve3nVzhWr845I7NynsOlcfFI=" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$x\\sqrt{36-x^{2}}:{\\quad}\\left(0,\\:0\\right)$$", "input": "x\\sqrt{36-x^{2}}", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=0\\sqrt{36-0^{2}}:{\\quad}y=0$$", "input": "y=0\\cdot\\:\\sqrt{36-0^{2}}", "steps": [ { "type": "interim", "title": "Simplify $$0\\cdot\\:\\sqrt{36-0^{2}}:{\\quad}0$$", "input": "0\\cdot\\:\\sqrt{36-0^{2}}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=0\\cdot\\:\\sqrt{36-0}" }, { "type": "interim", "title": "$$\\sqrt{36-0}=6$$", "input": "\\sqrt{36-0}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$36-0=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aEYi9wNujtaF0xOMmskYdVXTSum/z5kLpMzXS1UJIexyk0R8oZWybxT1ILwFJgqrhj6ylSwHjJkmQPKH34+7f385/YWN2Y6UOI+1UQSUpsQ=" } }, { "type": "step", "result": "=0\\cdot\\:6" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Y1o1ilPOV2x9Tl7301p1GMc9800WW+LP4/8QiFQJd8XNGoPE9TME3q+OPmgkv2RQz7FeFrf3rcM8IZlDz2c0dnql8XXPq6bNQlMm+36iNhlvxO1Phv2AeTzh2JMVpDP6P9r/UA07AB2VHNC0KM9RTmg5C2vkjV012+QEExcZEWY=" } }, { "type": "step", "result": "y=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfn/nW0uS5n29o1twgI0y74XWFYzcA4oaZR79v7JdxE/Fg5tGTrwPmIG+uvD8S0Rlxc3M8V4Y4QGRaO3chRVdsRU5QXr7DDZ9lFH7fhRgxCKI6FD+QELeWlZJF4zCe1SQWY=" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(0,\\:0\\right),\\:\\left(6,\\:0\\right),\\:\\left(-6,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x\\sqrt{36-x^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }