inflection x/(x^2+243)

{ "query": { "display": "inflection points $$\\frac{x}{x^{2}+243}$$", "symbolab_question": "FUNCTION#inflection \\frac{x}{x^{2}+243}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inflection", "default": "(-27,-\\frac{1}{36}),(0,0),(27,\\frac{1}{36})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inflection Points of $$\\frac{x}{x^{2}+243}:{\\quad}\\left(-27,\\:-\\frac{1}{36}\\right),\\:\\left(0,\\:0\\right),\\:\\left(27,\\:\\frac{1}{36}\\right)$$", "steps": [ { "type": "definition", "title": "Inflection points definition", "text": "An inflection point is a point on the graph at which the second derivative is equal to zero or undefined and changes sign.<br/>If $$f''\\left(x\\right)>0\\:$$then $$f\\left(x\\right)\\:$$concave upwards.<br/>If $$f''\\left(x\\right)<0\\:$$then $$f\\left(x\\right)\\:$$concave downwards." }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}$$", "input": "\\frac{d^{2}}{dx^{2}}\\left(\\frac{x}{x^{2}+243}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{x}{x^{2}+243}\\right)=\\frac{-x^{2}+243}{\\left(x^{2}+243\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{x}{x^{2}+243}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{dx}{dx}\\left(x^{2}+243\\right)-\\frac{d}{dx}\\left(x^{2}+243\\right)x}{\\left(x^{2}+243\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+243\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+243\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(243\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(243\\right)=0$$", "input": "\\frac{d}{dx}\\left(243\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYml4H/juFsxqoQxQtWNKm02XIQHgliMhSOSNsNni19Inf7LqB9CcyvYCWDsGseX09s1bIZxfodm3UsZcfZAZr4tQAFR5E8SBqowfht7doEgJJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1\\cdot\\:\\left(x^{2}+243\\right)-2xx}{\\left(x^{2}+243\\right)^{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+243\\right)-2xx=-x^{2}+243$$", "input": "1\\cdot\\:\\left(x^{2}+243\\right)-2xx", "result": "=\\frac{-x^{2}+243}{\\left(x^{2}+243\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+243\\right)=x^{2}+243$$", "input": "1\\cdot\\:\\left(x^{2}+243\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x^{2}+243\\right)=\\left(x^{2}+243\\right)$$", "result": "=\\left(x^{2}+243\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x^{2}+243" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXKw2WnVNECJh6R+TrH3aGOgdwkKGJWEPFPk38sdJMsyPIT8YedOUWtNErgRqsKwt0XD/5/7E/B7nZf9CiUamPankFubLRtQucVoygj65JVFtRjeYOMZL5j4OdhX/MBT08Jg==" } }, { "type": "interim", "title": "$$2xx=2x^{2}$$", "input": "2xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=2x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74UAakB14Lbm7SHYIpLTwTnCQoYlYQ8U+Tfyx0kyzI8iSVveXeWzQO/GlTVao5UKXszTt6qIJZczvODM49/dKgo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=x^{2}+243-2x^{2}" }, { "type": "step", "primary": "Group like terms", "result": "=x^{2}-2x^{2}+243" }, { "type": "step", "primary": "Add similar elements: $$x^{2}-2x^{2}=-x^{2}$$", "result": "=-x^{2}+243" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXK+HtypzHXR3bfLegfsBKW58DnzlbPZjyKgy1eUCFsLd5uuDaJPWwqUKQQbqdAE+F8D/L0MoYg+CUn6oyL3EO7YqmECoCSHsd3e9UcKjBCWhJx82fJkp3YFxoDefNaKDLQh6LVZdmyeEICJcpbdkKhko=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(\\frac{-x^{2}+243}{\\left(x^{2}+243\\right)^{2}}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{-x^{2}+243}{\\left(x^{2}+243\\right)^{2}}\\right)=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{-x^{2}+243}{\\left(x^{2}+243\\right)^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(-x^{2}+243\\right)\\left(x^{2}+243\\right)^{2}-\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)^{2}\\right)\\left(-x^{2}+243\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-x^{2}+243\\right)=-2x$$", "input": "\\frac{d}{dx}\\left(-x^{2}+243\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(243\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(243\\right)=0$$", "input": "\\frac{d}{dx}\\left(243\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYml4H/juFsxqoQxQtWNKm02XIQHgliMhSOSNsNni19Inf7LqB9CcyvYCWDsGseX09s1bIZxfodm3UsZcfZAZr4tQAFR5E8SBqowfht7doEgJJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-2x+0" }, { "type": "step", "primary": "Simplify", "result": "=-2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)^{2}\\right)=4x\\left(x^{2}+243\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)^{2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\left(x^{2}+243\\right)\\frac{d}{dx}\\left(x^{2}+243\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)^{2}\\right)", "result": "=2\\left(x^{2}+243\\right)\\frac{d}{dx}\\left(x^{2}+243\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\left(x^{2}+243\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dx}\\left(\\left(x^{2}+243\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(x^{2}+243\\right)$$", "result": "=2\\left(x^{2}+243\\right)\\frac{d}{dx}\\left(x^{2}+243\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpO/An25vI+ctBI+sjAuIMVMn/J4oSBqbI+AFzIHiA5BZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJUEQwPL90dVoLfth4U0tKl/+ABEAkV+3pW6PoqtAhfWaz8Bt2z2cDLOxZ/Hm1D3A0ohpPLyCYrLk9jd6X5FtSaWxnXWp25rOHIoAqY8LC2f/MJPkC5ONrC/KlB5u3a5uAAkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+243\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+243\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(243\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(243\\right)=0$$", "input": "\\frac{d}{dx}\\left(243\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYml4H/juFsxqoQxQtWNKm02XIQHgliMhSOSNsNni19Inf7LqB9CcyvYCWDsGseX09s1bIZxfodm3UsZcfZAZr4tQAFR5E8SBqowfht7doEgJJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2\\left(x^{2}+243\\right)\\cdot\\:2x" }, { "type": "step", "primary": "Simplify", "result": "=4x\\left(x^{2}+243\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\left(-2x\\right)\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-2x\\right)\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}:{\\quad}-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}$$", "input": "\\frac{\\left(-2x\\right)\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}", "result": "=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-2x\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "Factor $$-2x\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right):{\\quad}-2x\\left(x^{2}+243\\right)\\left(-x^{2}+729\\right)$$", "input": "-2x\\left(x^{2}+243\\right)^{2}-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)", "result": "=-\\frac{2x\\left(x^{2}+243\\right)\\left(-x^{2}+729\\right)}{\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\left(x^{2}+243\\right)^{2}=\\left(x^{2}+243\\right)\\left(x^{2}+243\\right)$$" ], "result": "=-2x\\left(x^{2}+243\\right)\\left(x^{2}+243\\right)-4x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite as", "result": "=-2x\\left(x^{2}+243\\right)\\left(x^{2}+243\\right)-2\\cdot\\:2x\\left(x^{2}+243\\right)\\left(-x^{2}+243\\right)" }, { "type": "step", "primary": "Factor out common term $$2x\\left(x^{2}+243\\right)$$", "result": "=-2x\\left(x^{2}+243\\right)\\left(x^{2}+243+2\\left(-x^{2}+243\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Expand $$x^{2}+2\\left(-x^{2}+243\\right)+243:{\\quad}-x^{2}+729$$", "input": "x^{2}+243+2\\left(-x^{2}+243\\right)", "result": "=-2x\\left(x^{2}+243\\right)\\left(-x^{2}+729\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\left(-x^{2}+243\\right):{\\quad}-2x^{2}+486$$", "input": "2\\left(-x^{2}+243\\right)", "result": "=x^{2}+243-2x^{2}+486", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=-x^{2},\\:c=243$$" ], "result": "=2\\left(-x^{2}\\right)+2\\cdot\\:243", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2x^{2}+2\\cdot\\:243" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:243=486$$", "result": "=-2x^{2}+486" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sLxpgW1Iotn1UfOTEw/CLy061ljBSPJeENOw2efoSWstl2HCB1voF4Kyr9fy3qDFPD/7lydZo+4buTe2ogeN0E3kCh3oevUunZ7/b0qFKBRgYxwiNcrruLAfiEMvAngciFyO5wp+FCCqC/ucNPDtpg==" } }, { "type": "interim", "title": "Simplify $$x^{2}+243-2x^{2}+486:{\\quad}-x^{2}+729$$", "input": "x^{2}+243-2x^{2}+486", "result": "=-x^{2}+729", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=x^{2}-2x^{2}+243+486" }, { "type": "step", "primary": "Add similar elements: $$x^{2}-2x^{2}=-x^{2}$$", "result": "=-x^{2}+243+486" }, { "type": "step", "primary": "Add the numbers: $$243+486=729$$", "result": "=-x^{2}+729" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qYuBHlhywOKyrZ8icK7958YWV3YT/KHjyh2PVLwqGjwDnzlbPZjyKgy1eUCFsLd5nL697FVQX2vCCE1TZMKVlD/L0MoYg+CUn6oyL3EO7Yqr4ZnuPqSW7yENxsmrnC1B2CTpNZ2CvsE9wWUejZXIPNTltPmNiR2fpmPA+5heJNg=" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}:{\\quad}\\left(x^{2}+243\\right)^{4}$$", "input": "\\left(\\left(x^{2}+243\\right)^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(x^{2}+243\\right)^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\left(x^{2}+243\\right)^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7h5Hai2m5OkELqLjIIguQyH0dPooFi9YZ7zCpVDwEMMsJQJZuTAY5js+oqjdT8kslqslxMZhmi5r3ycrrL9/IIvsicDtr1/4SZLlnwrW0smPuQCM/vqpbrqU5SxRHBPSduoCSgdt3GId0jvZgpS8aIRYnWVCZOo9IGqjQxeVReJs=" } }, { "type": "step", "result": "=-\\frac{2x\\left(x^{2}+243\\right)\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{4}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{2}+243$$", "result": "=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tHqt9JZCS3DFr3W5qRh8gy+WyfHm/cHfeQTbvWtiipXZR9eBEHQVmn9sKBO/EtXMGg87C5nSm7TBLnXy3qgp+Z0CRPuafT39sUtXUeZie3ktOtZYwUjyXhDTsNnn6ElryOsg4xTbsj8PJfnagYu7Q+KlKTJJ+yfcPmWGr5C4MJdIEco24GJ2gC0lvje8XpN00zMYqEwpcuv4TrlApQ7eppjcBIL5pmo83UMFZRSzJsWSeLnzvuRdJUznWVm62XjccVKD2v15JLttiu7+Gkl/pG6KMHR13W3SUAX7T7jnacjxPcfbN2hXY0hcdjnG4NRVOTqg4xGSQir3s9SBkzzdDliVI3uvN1by+AN9NfjoKFU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Concave Downward$$:-\\infty\\:<x<-27,\\:$$Concave Upward$$:-27<x<0,\\:$$Concave Downward$$:0<x<27,\\:$$Concave Upward$$:27<x<\\infty\\:$$", "input": "f\\:{^{\\prime\\prime}}\\left(x\\right)=-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}", "steps": [ { "type": "interim", "title": "Find where $$f\\:{^{\\prime\\prime}}\\left(x\\right)$$ is equal to zero or undefined:$${\\quad}x=-27,\\:x=0,\\:x=27$$", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=0:{\\quad}x=0,\\:x=-27,\\:x=27$$", "input": "-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "2x\\left(-x^{2}+729\\right)=0" }, { "type": "interim", "title": "Solve $$2x\\left(-x^{2}+729\\right)=0:{\\quad}x=0,\\:x=-27,\\:x=27$$", "input": "2x\\left(-x^{2}+729\\right)=0", "steps": [ { "type": "interim", "title": "Factor $$2x\\left(-x^{2}+729\\right):{\\quad}-2x\\left(x+27\\right)\\left(x-27\\right)$$", "input": "2x\\left(-x^{2}+729\\right)", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+729:{\\quad}-\\left(x+27\\right)\\left(x-27\\right)$$", "input": "-x^{2}+729", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-729\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-729:{\\quad}\\left(x+27\\right)\\left(x-27\\right)$$", "input": "x^{2}-729", "steps": [ { "type": "step", "primary": "Rewrite $$729$$ as $$27^{2}$$", "result": "=x^{2}-27^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-27^{2}=\\left(x+27\\right)\\left(x-27\\right)$$" ], "result": "=\\left(x+27\\right)\\left(x-27\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-2x\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "-2x\\left(x+27\\right)\\left(x-27\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x=0\\lor\\:x+27=0\\lor\\:x-27=0" }, { "type": "interim", "title": "Solve $$x+27=0:{\\quad}x=-27$$", "input": "x+27=0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x+27=0", "result": "x=-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27=0-27" }, { "type": "step", "primary": "Simplify", "result": "x=-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x-27=0:{\\quad}x=27$$", "input": "x-27=0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x-27=0", "result": "x=27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "x-27+27=0+27" }, { "type": "step", "primary": "Simplify", "result": "x=27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "x=0,\\:x=-27,\\:x=27" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=0,\\:x=-27,\\:x=27" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=-27,\\:x=0,\\:x=27" } ], "meta": { "interimType": "Explore Function Slope Zero Specific 1Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)>0:{\\quad}-27<x<0\\lor\\:x>27$$", "input": "-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "result": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}\\right)\\left(-1\\right)<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}}{2}<\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}<0" } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}:{\\quad}\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}$$", "input": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}", "result": "\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "steps": [ { "type": "interim", "title": "Factor $$x\\left(-x^{2}+729\\right):{\\quad}-x\\left(x+27\\right)\\left(x-27\\right)$$", "input": "x\\left(-x^{2}+729\\right)", "result": "=\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+729:{\\quad}-\\left(x+27\\right)\\left(x-27\\right)$$", "input": "-x^{2}+729", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-729\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-729:{\\quad}\\left(x+27\\right)\\left(x-27\\right)$$", "input": "x^{2}-729", "steps": [ { "type": "step", "primary": "Rewrite $$729$$ as $$27^{2}$$", "result": "=x^{2}-27^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-27^{2}=\\left(x+27\\right)\\left(x-27\\right)$$" ], "result": "=\\left(x+27\\right)\\left(x-27\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-x\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-x\\left(x+27\\right)\\left(x-27\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+243\\right)^{3}}>0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "-27<x<0\\lor\\:x>27", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}$$" }, { "type": "interim", "title": "Find the signs of $$x$$", "steps": [ { "type": "step", "result": "x=0" }, { "type": "step", "result": "x<0" }, { "type": "step", "result": "x>0" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x+27$$", "steps": [ { "type": "interim", "title": "$$x+27=0:{\\quad}x=-27$$", "input": "x+27=0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x+27=0", "result": "x=-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27=0-27" }, { "type": "step", "primary": "Simplify", "result": "x=-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x+27<0:{\\quad}x<-27$$", "input": "x+27<0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x+27<0", "result": "x<-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27<0-27" }, { "type": "step", "primary": "Simplify", "result": "x<-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x+27>0:{\\quad}x>-27$$", "input": "x+27>0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x+27>0", "result": "x>-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27>0-27" }, { "type": "step", "primary": "Simplify", "result": "x>-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "x<27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "x-27+27<0+27" }, { "type": "step", "primary": "Simplify", "result": "x<27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-27>0:{\\quad}x>27$$", "input": "x-27>0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x-27>0", "result": "x>27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "x-27+27>0+27" }, { "type": "step", "primary": "Simplify", "result": "x>27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(x^{2}+243\\right)^{3}:{\\quad}$$No Solution", "input": "\\left(x^{2}+243\\right)^{3}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+243=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+243=0", "steps": [ { "type": "interim", "title": "Move $$243\\:$$to the right side", "input": "x^{2}+243=0", "result": "x^{2}=-243", "steps": [ { "type": "step", "primary": "Subtract $$243$$ from both sides", "result": "x^{2}+243-243=0-243" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-243" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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Title 1Eq" } }, { "type": "step", "primary": "The solution is" }, { "type": "step", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|c|c|}\\hline &x<-27&x=-27&-27<x<0&x=0&0<x<27&x=27&x>27\\\\\\hline x&-&-&-&0&+&+&+\\\\\\hline x+27&-&0&+&+&+&+&+\\\\\\hline x-27&-&-&-&-&-&0&+\\\\\\hline (x^{2}+243)^{3}&+&+&+&+&+&+&+\\\\\\hline \\frac{x(x+27)(x-27)}{(x^{2}+243)^{3}}&-&0&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "-27<x<0\\lor\\:x>27" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)<0:{\\quad}x<-27\\lor\\:0<x<27$$", "input": "-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "result": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}\\right)\\left(-1\\right)>0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{\\frac{2x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}}{2}>\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}>0" } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}:{\\quad}\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}$$", "input": "\\frac{x\\left(-x^{2}+729\\right)}{\\left(x^{2}+243\\right)^{3}}", "result": "\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}>0", "steps": [ { "type": "interim", "title": "Factor $$x\\left(-x^{2}+729\\right):{\\quad}-x\\left(x+27\\right)\\left(x-27\\right)$$", "input": "x\\left(-x^{2}+729\\right)", "result": "=\\frac{-x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+729:{\\quad}-\\left(x+27\\right)\\left(x-27\\right)$$", "input": "-x^{2}+729", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-729\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-729:{\\quad}\\left(x+27\\right)\\left(x-27\\right)$$", "input": "x^{2}-729", "steps": [ { "type": "step", "primary": "Rewrite $$729$$ as $$27^{2}$$", "result": "=x^{2}-27^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-27^{2}=\\left(x+27\\right)\\left(x-27\\right)$$" ], "result": "=\\left(x+27\\right)\\left(x-27\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-x\\left(x+27\\right)\\left(x-27\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-x\\left(x+27\\right)\\left(x-27\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+243\\right)^{3}}<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}<0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "x<-27\\lor\\:0<x<27", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{x\\left(x+27\\right)\\left(x-27\\right)}{\\left(x^{2}+243\\right)^{3}}$$" }, { "type": "interim", "title": "Find the signs of $$x$$", "steps": [ { "type": "step", "result": "x=0" }, { "type": "step", "result": "x<0" }, { "type": "step", "result": "x>0" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x+27$$", "steps": [ { "type": "interim", "title": "$$x+27=0:{\\quad}x=-27$$", "input": "x+27=0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x+27=0", "result": "x=-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27=0-27" }, { "type": "step", "primary": "Simplify", "result": "x=-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "x<-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27<0-27" }, { "type": "step", "primary": "Simplify", "result": "x<-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"x+27>0", "result": "x>-27", "steps": [ { "type": "step", "primary": "Subtract $$27$$ from both sides", "result": "x+27-27>0-27" }, { "type": "step", "primary": "Simplify", "result": "x>-27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "x<27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "x-27+27<0+27" }, { "type": "step", "primary": "Simplify", "result": "x<27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oDZx10hZX1pxONbvtDpzthyOUtErWNRXbTqqjj69E3s34A/w/mRw9In1azMIY2eLPWotfZYazYMcdbv0a0nyV+mMBETnBNKfRtyTzXttszW8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAyUOxcOPJc9ZnUJOTEXqkBo2iqHCtSte9VzI0e+Pw+g2lGiO4MK5XQQNAqG2dGAKJ6wi29LxZs728xgYz2JweTh2wjqcGr0codC9/hhcMsvfeEz0lqgO2FuFHQomhmczQwbkkRES0A/nVu8HXpy3IvDksg1aC97qvSO80ZinAIXC6brd1/tYQ2W+fdNKDawSZFAFM2L0DNZ9bwH1OnQcskVJxPj25EAfGWqVDcg4Tx9uFMLot0MDFxYg7oznHcQL7gZXo/GjHUKwJqnFPRERdJxlnenT7OF2d9W75/QEKjqVzfiguftzSeHNczvBup8dIhZ9VXfbSWYapymUWw0lt8JNJFpieUqwSM10mklZi3jyKaaZ8ef2sav2E0ahdMP64rO6Zx1D8371h76W0d6Z98dGdF4kg2/TXlR60HycP62pWG2POe4RJGW6hHHywAfGSwbO5LfCk5wxQy4uSVeWSNTgOhFcKXKNhVNjhBHfbnqH/BgeR+YcdPN7cleTcuOImk=" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-27>0:{\\quad}x>27$$", "input": "x-27>0", "steps": [ { "type": "interim", "title": "Move $$27\\:$$to the right side", "input": "x-27>0", "result": "x>27", "steps": [ { "type": "step", "primary": "Add $$27$$ to both sides", "result": "x-27+27>0+27" }, { "type": "step", "primary": "Simplify", "result": "x>27" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(x^{2}+243\\right)^{3}:{\\quad}$$No Solution", "input": "\\left(x^{2}+243\\right)^{3}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+243=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+243=0", "steps": [ { "type": "interim", "title": "Move $$243\\:$$to the right side", "input": "x^{2}+243=0", "result": "x^{2}=-243", "steps": [ { "type": "step", "primary": "Subtract $$243$$ from both sides", "result": "x^{2}+243-243=0-243" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-243" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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Title 1Eq" } }, { "type": "step", "primary": "The solution is" }, { "type": "step", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|c|c|}\\hline &x<-27&x=-27&-27<x<0&x=0&0<x<27&x=27&x>27\\\\\\hline x&-&-&-&0&+&+&+\\\\\\hline x+27&-&0&+&+&+&+&+\\\\\\hline x-27&-&-&-&-&-&0&+\\\\\\hline (x^{2}+243)^{3}&+&+&+&+&+&+&+\\\\\\hline \\frac{x(x+27)(x-27)}{(x^{2}+243)^{3}}&-&0&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$<\\:0$$", "result": "x<-27\\lor\\:0<x<27" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<-27,\\:x=-27,\\:-27<x<0,\\:x=0,\\:0<x<27,\\:x=27,\\:27<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$\\frac{x}{x^{2}+243}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Combine $$x=-27\\:$$ with domain:$${\\quad}x=-27$$", "input": "x=-27\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=-27" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$x=0\\:$$ with domain:$${\\quad}x=0$$", "input": "x=0\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$x=27\\:$$ with domain:$${\\quad}x=27$$", "input": "x=27\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=27" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-27<x<0\\:$$ with domain:$${\\quad}-27<x<0$$", "input": "-27<x<0\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-27<x<0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$27<x<\\infty\\:\\:$$ with domain:$${\\quad}27<x<\\infty\\:$$", "input": "27<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "27<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<-27\\:$$ with domain:$${\\quad}-\\infty\\:<x<-27$$", "input": "-\\infty\\:<x<-27\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<-27" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$0<x<27\\:$$ with domain:$${\\quad}0<x<27$$", "input": "0<x<27\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "0<x<27" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<-27,\\:x=-27,\\:-27<x<0,\\:x=0,\\:0<x<27,\\:x=27,\\:27<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the sign intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|c|c|}\\hline &-\\infty <x<-27&x=-27&-27<x<0&x=0&0<x<27&x=27&27<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime\\prime}}(x)<0&f {^{\\prime\\prime}}(x)=0&f {^{\\prime\\prime}}(x)>0&f {^{\\prime\\prime}}(x)=0&f {^{\\prime\\prime}}(x)<0&f {^{\\prime\\prime}}(x)=0&f {^{\\prime\\prime}}(x)>0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Concave\\:Downward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Upward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Downward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Upward}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Concave\\:Downward}:-\\infty\\:<x<-27,\\:\\mathrm{Concave\\:Upward}:-27<x<0,\\:\\mathrm{Concave\\:Downward}:0<x<27,\\:\\mathrm{Concave\\:Upward}:27<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQWS/fnqlAfkusYvPtw7A/ttoqhwrUrXvVcyNHvj8PoNpRojuDCuV0EDQKhtnRgCiesItvS8WbO9vMYGM9icHk4T2k7cCzkNVN/aeeCLNiqEi7Dfz4tvuOcJYf1t7mfg8LrC+kc6fU1DBatrQQT4hsMA==" } }, { "type": "interim", "title": "Plug $$x=-27\\:$$into $$\\frac{x}{x^{2}+243}:{\\quad}-\\frac{1}{36}$$", "input": "\\frac{\\left(-27\\right)}{\\left(-27\\right)^{2}+243}", "result": "\\left(-27,\\:-\\frac{1}{36}\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\frac{1}{36}" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cfvukFYDWdb0TVi911tLQW7i4dlyWzar2iP8FF/i/AFBKZwi/MOEiiRh/mdNlF9Kx5EYTybQyDBefJ06k4AZm8P5sshoSXqVvJSK2LqsBmVTwTC5ZvDXrRuKgad4NesZxmf4Sa+Rvy4nbg5ayx7itYQ==" } }, { "type": "interim", "title": "Plug $$x=0\\:$$into $$\\frac{x}{x^{2}+243}:{\\quad}0$$", "input": "\\frac{0}{0^{2}+243}", "result": "\\left(0,\\:0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "0" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7ccA1hpeCt0rRiP9iJbxBbzfmADZxgltOL0+sjlMf3bblkS3dlcCKpQTQcheuut7MkS4X77HvSt7slSZWQ9ahxI8Ylnswdhi24TCGTiq7SjJ1AFM2L0DNZ9bwH1OnQcskVAWCs/EBTzTiukzvjGN3gJw==" } }, { "type": "interim", "title": "Plug $$x=27\\:$$into 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