inflection (x^2+x+1)/(x^2-x+1)

{ "query": { "display": "inflection points $$\\frac{x^{2}+x+1}{x^{2}-x+1}$$", "symbolab_question": "FUNCTION#inflection \\frac{x^{2}+x+1}{x^{2}-x+1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inflection", "default": "(-1.87938…,0.41374…),(0.34729…,1.89819…),(1.53208…,2.68805…)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inflection Points of $$\\frac{x^{2}+x+1}{x^{2}-x+1}:{\\quad}\\left(-1.87938…,\\:0.41374…\\right),\\:\\left(0.34729…,\\:1.89819…\\right),\\:\\left(1.53208…,\\:2.68805…\\right)$$", "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": "Find where $$f^{\\prime\\prime}\\left(x\\right)$$ is equal to zero or undefined", "input": "\\frac{x^{2}+x+1}{x^{2}-x+1}", "result": "x=-1.87938…,\\:x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…", "steps": [ { "type": "interim", "title": "$$f^{\\prime\\prime}\\left(x\\right)=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "input": "\\frac{d^{2}}{dx^{2}}\\left(\\frac{x^{2}+x+1}{x^{2}-x+1}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{x^{2}+x+1}{x^{2}-x+1}\\right)=\\frac{-2x^{2}+2}{\\left(x^{2}-x+1\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{x^{2}+x+1}{x^{2}-x+1}\\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}+x+1\\right)\\left(x^{2}-x+1\\right)-\\frac{d}{dx}\\left(x^{2}-x+1\\right)\\left(x^{2}+x+1\\right)}{\\left(x^{2}-x+1\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+x+1\\right)=2x+1$$", "input": "\\frac{d}{dx}\\left(x^{2}+x+1\\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{dx}{dx}+\\frac{d}{dx}\\left(1\\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": "$$\\frac{dx}{dx}=1$$", "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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x+1+0" }, { "type": "step", "primary": "Simplify", "result": "=2x+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-x+1\\right)=2x-1$$", "input": "\\frac{d}{dx}\\left(x^{2}-x+1\\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{dx}{dx}+\\frac{d}{dx}\\left(1\\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": "$$\\frac{dx}{dx}=1$$", "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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x-1+0" }, { "type": "step", "primary": "Simplify", "result": "=2x-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\left(2x+1\\right)\\left(x^{2}-x+1\\right)-\\left(2x-1\\right)\\left(x^{2}+x+1\\right)}{\\left(x^{2}-x+1\\right)^{2}}" }, { "type": "interim", "title": "Expand $$\\left(2x+1\\right)\\left(x^{2}-x+1\\right)-\\left(2x-1\\right)\\left(x^{2}+x+1\\right):{\\quad}-2x^{2}+2$$", "input": "\\left(2x+1\\right)\\left(x^{2}-x+1\\right)-\\left(2x-1\\right)\\left(x^{2}+x+1\\right)", "result": "=\\frac{-2x^{2}+2}{\\left(x^{2}-x+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "Expand $$\\left(2x+1\\right)\\left(x^{2}-x+1\\right):{\\quad}2x^{3}-x^{2}+x+1$$", "input": "\\left(2x+1\\right)\\left(x^{2}-x+1\\right)", "result": "=2x^{3}-x^{2}+x+1-\\left(2x-1\\right)\\left(x^{2}+x+1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=2xx^{2}+2x\\left(-x\\right)+2x\\cdot\\:1+1\\cdot\\:x^{2}+1\\cdot\\:\\left(-x\\right)+1\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the first parentheses<br/>by each of the terms within the second parentheses left to right" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=2x^{2}x-2xx+2\\cdot\\:1\\cdot\\:x+1\\cdot\\:x^{2}-1\\cdot\\:x+1\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$2x^{2}x-2xx+2\\cdot\\:1\\cdot\\:x+1\\cdot\\:x^{2}-1\\cdot\\:x+1\\cdot\\:1:{\\quad}2x^{3}-x^{2}+x+1$$", "input": "2x^{2}x-2xx+2\\cdot\\:1\\cdot\\:x+1\\cdot\\:x^{2}-1\\cdot\\:x+1\\cdot\\:1", "result": "=2x^{3}-x^{2}+x+1", "steps": [ { "type": "interim", "title": "$$2x^{2}x=2x^{3}$$", "input": "2x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=2x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70thjYFL6CU6+q37zoPyD/wOfOVs9mPIqDLV5QIWwt3mwB/QJ3d78LroQvy/1JTpnx06diEhTBX3c/BKR15lU3GMU8hJEL7k4ZGKY8HFIubc=" } }, { "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": "interim", "title": "$$2\\cdot\\:1\\cdot\\:x=2x$$", "input": "2\\cdot\\:1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFJrzCAnUlzDDpdpijAydC16jkVi15I8rBefLi4Iyt2wr2GIoxg2Jpr0LaZPQ02JWI3uMoyubzblmmWXBbGmpBNCOUevHcCys/ACQReKIPyPr" } }, { "type": "interim", "title": "$$1\\cdot\\:x^{2}=x^{2}$$", "input": "1\\cdot\\:x^{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78hVycPB+rfQCHnyh7m3HTi061ljBSPJeENOw2efoSWt5uKsWh4cBdpi/wldLLf2V/z//r+dXk7h9vxeDCLuZqsWPXBkUr6zzMsua0zkIRDrF3DenEWojLSGXYMDcAl7b" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "interim", "title": "$$1\\cdot\\:1=1$$", "input": "1\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AqkzxZA9V4q7Df6awsBYU913jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fSPJrYhwc+zvuHrOLz58Ml2lcUv7BL7DC3vHXcXDfb5KE=" } }, { "type": "step", "result": "=2x^{3}-2x^{2}+2x+x^{2}-x+1" }, { "type": "step", "primary": "Group like terms", "result": "=2x^{3}-2x^{2}+x^{2}+2x-x+1" }, { "type": "step", "primary": "Add similar elements: $$-2x^{2}+x^{2}=-x^{2}$$", "result": "=2x^{3}-x^{2}+2x-x+1" }, { "type": "step", "primary": "Add similar elements: $$2x-x=x$$", "result": "=2x^{3}-x^{2}+x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+80lZwmUmg/Pz8eId9FGvT2gYtQEl4V79ys4pDeUXjWjkVi15I8rBefLi4Iyt2wrNEUa1NAwqfV0bDKqcv2BLT/L0MoYg+CUn6oyL3EO7Yqr4ZnuPqSW7yENxsmrnC1BkXQhGFTNi7MJVg/a0hvN6WHE33NZtV6QHzGTclQ6fVo=" } }, { "type": "interim", "title": "Expand $$-\\left(2x-1\\right)\\left(x^{2}+x+1\\right):{\\quad}-2x^{3}-x^{2}-x+1$$", "result": "=2x^{3}-x^{2}+x+1-2x^{3}-x^{2}-x+1", "steps": [ { "type": "interim", "title": "Expand $$\\left(2x-1\\right)\\left(x^{2}+x+1\\right):{\\quad}2x^{3}+x^{2}+x-1$$", "input": "\\left(2x-1\\right)\\left(x^{2}+x+1\\right)", "result": "=-\\left(2x^{3}+x^{2}+x-1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=2xx^{2}+2xx+2x\\cdot\\:1+\\left(-1\\right)x^{2}+\\left(-1\\right)x+\\left(-1\\right)\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the first parentheses<br/>by each of the terms within the second parentheses left to right" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=2x^{2}x+2xx+2\\cdot\\:1\\cdot\\:x-1\\cdot\\:x^{2}-1\\cdot\\:x-1\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$2x^{2}x+2xx+2\\cdot\\:1\\cdot\\:x-1\\cdot\\:x^{2}-1\\cdot\\:x-1\\cdot\\:1:{\\quad}2x^{3}+x^{2}+x-1$$", "input": "2x^{2}x+2xx+2\\cdot\\:1\\cdot\\:x-1\\cdot\\:x^{2}-1\\cdot\\:x-1\\cdot\\:1", "result": "=2x^{3}+x^{2}+x-1", "steps": [ { "type": "interim", "title": "$$2x^{2}x=2x^{3}$$", "input": "2x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=2x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70thjYFL6CU6+q37zoPyD/wOfOVs9mPIqDLV5QIWwt3mwB/QJ3d78LroQvy/1JTpnx06diEhTBX3c/BKR15lU3GMU8hJEL7k4ZGKY8HFIubc=" } }, { "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": "interim", "title": "$$2\\cdot\\:1\\cdot\\:x=2x$$", "input": "2\\cdot\\:1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFJrzCAnUlzDDpdpijAydC16jkVi15I8rBefLi4Iyt2wr2GIoxg2Jpr0LaZPQ02JWI3uMoyubzblmmWXBbGmpBNCOUevHcCys/ACQReKIPyPr" } }, { "type": "interim", "title": "$$1\\cdot\\:x^{2}=x^{2}$$", "input": "1\\cdot\\:x^{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78hVycPB+rfQCHnyh7m3HTi061ljBSPJeENOw2efoSWt5uKsWh4cBdpi/wldLLf2V/z//r+dXk7h9vxeDCLuZqsWPXBkUr6zzMsua0zkIRDrF3DenEWojLSGXYMDcAl7b" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "interim", "title": "$$1\\cdot\\:1=1$$", "input": "1\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AqkzxZA9V4q7Df6awsBYU913jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fSPJrYhwc+zvuHrOLz58Ml2lcUv7BL7DC3vHXcXDfb5KE=" } }, { "type": "step", "result": "=2x^{3}+2x^{2}+2x-x^{2}-x-1" }, { "type": "step", "primary": "Group like terms", "result": "=2x^{3}+2x^{2}-x^{2}+2x-x-1" }, { "type": "step", "primary": "Add similar elements: $$2x^{2}-x^{2}=x^{2}$$", "result": "=2x^{3}+x^{2}+2x-x-1" }, { "type": "step", "primary": "Add similar elements: $$2x-x=x$$", "result": "=2x^{3}+x^{2}+x-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CFsi8CFBMUuKNZ5qElh9dj2gYtQEl4V79ys4pDeUXjWjkVi15I8rBefLi4Iyt2wrwN4cFhuctnLzxPkoG1K/XD/L0MoYg+CUn6oyL3EO7Yqr4ZnuPqSW7yENxsmrnC1B88PRAgvSn7Jlmz26Ab7Hb2HE33NZtV6QHzGTclQ6fVo=" } }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(2x^{3}\\right)-\\left(x^{2}\\right)-\\left(x\\right)-\\left(-1\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-2x^{3}-x^{2}-x+1" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq" } }, { "type": "interim", "title": "Simplify $$2x^{3}-x^{2}+x+1-2x^{3}-x^{2}-x+1:{\\quad}-2x^{2}+2$$", "input": "2x^{3}-x^{2}+x+1-2x^{3}-x^{2}-x+1", "result": "=-2x^{2}+2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2x^{3}-2x^{3}-x^{2}-x^{2}+x-x+1+1" }, { "type": "step", "primary": "Add similar elements: $$-x^{2}-x^{2}=-2x^{2}$$", "result": "=2x^{3}-2x^{3}-2x^{2}+x-x+1+1" }, { "type": "step", "primary": "Add similar elements: $$2x^{3}-2x^{3}=0$$", "result": "=-2x^{2}+x-x+1+1" }, { "type": "step", "primary": "Add similar elements: $$x-x=0$$", "result": "=-2x^{2}+1+1" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=-2x^{2}+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+80lZwmUmg/Pz8eId9FGva57U9eSOmyocaiJJ8j0+ntF+ulRTPtbdJ9lB1KElXHTcJChiVhDxT5N/LHSTLMjyPKOyxhisjdl9l8QPPd5t1iBBTEk/JQ2cZ9WKuRzClU7eahtdqtmUkkRVWanKrj0lvLNesSUZ3PRKKDS3BxwJtUA6QaMRfKMrDZ7B3LuBozVvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(\\frac{-2x^{2}+2}{\\left(x^{2}-x+1\\right)^{2}}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{-2x^{2}+2}{\\left(x^{2}-x+1\\right)^{2}}\\right)=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{-2x^{2}+2}{\\left(x^{2}-x+1\\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(-2x^{2}+2\\right)\\left(x^{2}-x+1\\right)^{2}-\\frac{d}{dx}\\left(\\left(x^{2}-x+1\\right)^{2}\\right)\\left(-2x^{2}+2\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-2x^{2}+2\\right)=-4x$$", "input": "\\frac{d}{dx}\\left(-2x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(2x^{2}\\right)+\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x^{2}\\right)=4x$$", "input": "\\frac{d}{dx}\\left(2x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgEShBvy1snibnAN3NvB1a2TdaV09PMxEKZ9FieghTFwHBO3D9VaGp1eOVvjTiCiEaN6Hv6MoTMtvtU0IQwXdn+XNwOQ43NHE8cpERrPgoqpfTZuddhTh3r/FmyVu4x1Bw==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(2\\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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "step", "result": "=-4x+0" }, { "type": "step", "primary": "Simplify", "result": "=-4x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\left(x^{2}-x+1\\right)^{2}\\right)=2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}-x+1\\right)^{2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\left(x^{2}-x+1\\right)\\frac{d}{dx}\\left(x^{2}-x+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}-x+1\\right)^{2}\\right)", "result": "=2\\left(x^{2}-x+1\\right)\\frac{d}{dx}\\left(x^{2}-x+1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\left(x^{2}-x+1\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dx}\\left(\\left(x^{2}-x+1\\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}-x+1\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(x^{2}-x+1\\right)$$", "result": "=2\\left(x^{2}-x+1\\right)\\frac{d}{dx}\\left(x^{2}-x+1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsxid4ZFpbQoV+tO1E6uamU57tQX3BEe2xWeTm8uV1kpZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJUEQwPL90dVoLfth4U0tKl/9qVzXEI6tyQmuTus0RzXScGwfWJyr5JWzRKO1dICJXohpPLyCYrLk9jd6X5FtSaWxnXWp25rOHIoAqY8LC2f/MJPkC5ONrC/KlB5u3a5uAAkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-x+1\\right)=2x-1$$", "input": "\\frac{d}{dx}\\left(x^{2}-x+1\\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{dx}{dx}+\\frac{d}{dx}\\left(1\\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": "$$\\frac{dx}{dx}=1$$", "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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x-1+0" }, { "type": "step", "primary": "Simplify", "result": "=2x-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\left(-4x\\right)\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(-4x\\right)\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}:{\\quad}-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "input": "\\frac{\\left(-4x\\right)\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}", "result": "=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-4x\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}" }, { "type": "interim", "title": "Factor $$-4x\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right):{\\quad}-4\\left(x^{2}+1-x\\right)\\left(-x^{3}+3x-1\\right)$$", "input": "-4x\\left(x^{2}-x+1\\right)^{2}-2\\left(x^{2}-x+1\\right)\\left(2x-1\\right)\\left(-2x^{2}+2\\right)", "result": "=-\\frac{4\\left(x^{2}+1-x\\right)\\left(-x^{3}+3x-1\\right)}{\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\left(x^{2}+1-x\\right)^{2}=\\left(x^{2}+1-x\\right)\\left(x^{2}+1-x\\right)$$" ], "result": "=-4x\\left(x^{2}+1-x\\right)\\left(x^{2}+1-x\\right)-2\\left(x^{2}+1-x\\right)\\left(-1+2x\\right)\\left(2-x^{2}\\cdot\\:2\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite as", "result": "=-2\\cdot\\:2\\left(x^{2}+1-x\\right)x\\left(x^{2}+1-x\\right)-2\\left(x^{2}+1-x\\right)\\left(-1+2x\\right)\\left(2-x^{2}\\cdot\\:2\\right)" }, { "type": "step", "primary": "Factor out common term $$2\\left(x^{2}+1-x\\right)$$", "result": "=-2\\left(x^{2}+1-x\\right)\\left(2x\\left(x^{2}+1-x\\right)+\\left(-1+2x\\right)\\left(2-x^{2}\\cdot\\:2\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$2x\\left(x^{2}-x+1\\right)+\\left(2x-1\\right)\\left(-2x^{2}+2\\right):{\\quad}2\\left(-x^{3}+3x-1\\right)$$", "input": "2x\\left(x^{2}+1-x\\right)+\\left(-1+2x\\right)\\left(2-x^{2}\\cdot\\:2\\right)", "result": "=-2\\cdot\\:2\\left(x^{2}-x+1\\right)\\left(-x^{3}+3x-1\\right)", "steps": [ { "type": "interim", "title": "Factor $$2-x^{2}2:{\\quad}2\\left(1-x^{2}\\right)$$", "input": "2-x^{2}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:1-2x^{2}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(1-x^{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=2x\\left(x^{2}-x+1\\right)+2\\left(2x-1\\right)\\left(-x^{2}+1\\right)" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x\\left(x^{2}+1-x\\right)+\\left(-1+2x\\right)\\left(-x^{2}+1\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Expand $$x\\left(x^{2}-x+1\\right)+\\left(2x-1\\right)\\left(-x^{2}+1\\right):{\\quad}-x^{3}+3x-1$$", "input": "x\\left(x^{2}+1-x\\right)+\\left(-1+2x\\right)\\left(-x^{2}+1\\right)", "result": "=2\\left(-x^{3}+3x-1\\right)", "steps": [ { "type": "interim", "title": "Expand $$x\\left(x^{2}+1-x\\right):{\\quad}x^{3}+x-x^{2}$$", "input": "x\\left(x^{2}+1-x\\right)", "result": "=x^{3}+x-x^{2}+\\left(-1+2x\\right)\\left(-x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=xx^{2}+x\\cdot\\:1+x\\left(-x\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=x^{2}x+1\\cdot\\:x-xx" }, { "type": "interim", "title": "Simplify $$x^{2}x+1\\cdot\\:x-xx:{\\quad}x^{3}+x-x^{2}$$", "input": "x^{2}x+1\\cdot\\:x-xx", "result": "=x^{3}+x-x^{2}", "steps": [ { "type": "interim", "title": "$$x^{2}x=x^{3}$$", "input": "x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LWrIAcT/Pu2l3a2UlrsCC3WD310L1+P2yDQQfMEhENGDPbrpZaMLRegCZc+JnvJI5kmDAAHjIPJcICsCIhoRbVuSVZd9z4+kRKtqsjU2P18=" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "interim", "title": "$$xx=x^{2}$$", "input": "xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74tglIeOAQqOR+OcPiyzQUczBWJotReR4P4m6RE6FZ2M7Aq6fHyeqJtW5OKbXVcT+IBF/biSmVq3Z2pV/8nBrAiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=x^{3}+x-x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75tA6/vu9iUKpax8S63wQCVXTSum/z5kLpMzXS1UJIeyzxKcFfLfF1kW+O2Jhuez24KtEDIv3hNWTFAKyRUzq43ql8XXPq6bNQlMm+36iNhmwxxgL38iSshvJ79iO6+ditC3lhDE6uOxDQHuaXj5OMA==" } }, { "type": "interim", "title": "Expand $$\\left(-1+2x\\right)\\left(-x^{2}+1\\right):{\\quad}x^{2}-1-2x^{3}+2x$$", "input": "\\left(-1+2x\\right)\\left(-x^{2}+1\\right)", "result": "=x^{3}+x-x^{2}+x^{2}-1-2x^{3}+2x", "steps": [ { "type": "step", "primary": "Apply FOIL method: $$\\left(a+b\\right)\\left(c+d\\right)=ac+ad+bc+bd$$", "secondary": [ "$$a=-1,\\:b=2x,\\:c=-x^{2},\\:d=1$$" ], "result": "=\\left(-1\\right)\\left(-x^{2}\\right)+\\left(-1\\right)\\cdot\\:1+2x\\left(-x^{2}\\right)+2x\\cdot\\:1", "meta": { "title": { "extension": "F-First<br/>O-Outer<br/>I-Inner<br/>L-Last" }, "practiceLink": "/practice/expansion-practice#area=main&subtopic=FOIL_Basic", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$\\left(-a\\right)\\left(-b\\right)=ab,\\:\\:+\\left(-a\\right)=-a$$" ], "result": "=1\\cdot\\:x^{2}-1\\cdot\\:1-2x^{2}x+2\\cdot\\:1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:x^{2}-1\\cdot\\:1-2x^{2}x+2\\cdot\\:1\\cdot\\:x:{\\quad}x^{2}-1-2x^{3}+2x$$", "input": "1\\cdot\\:x^{2}-1\\cdot\\:1-2x^{2}x+2\\cdot\\:1\\cdot\\:x", "result": "=x^{2}-1-2x^{3}+2x", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:x^{2}=x^{2}$$", "input": "1\\cdot\\:x^{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78hVycPB+rfQCHnyh7m3HTi061ljBSPJeENOw2efoSWt5uKsWh4cBdpi/wldLLf2V/z//r+dXk7h9vxeDCLuZqsWPXBkUr6zzMsua0zkIRDrF3DenEWojLSGXYMDcAl7b" } }, { "type": "interim", "title": "$$1\\cdot\\:1=1$$", "input": "1\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AqkzxZA9V4q7Df6awsBYU913jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fSPJrYhwc+zvuHrOLz58Ml2lcUv7BL7DC3vHXcXDfb5KE=" } }, { "type": "interim", "title": "$$2x^{2}x=2x^{3}$$", "input": "2x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=2x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70thjYFL6CU6+q37zoPyD/wOfOVs9mPIqDLV5QIWwt3mwB/QJ3d78LroQvy/1JTpnx06diEhTBX3c/BKR15lU3GMU8hJEL7k4ZGKY8HFIubc=" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:x=2x$$", "input": "2\\cdot\\:1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFJrzCAnUlzDDpdpijAydC16jkVi15I8rBefLi4Iyt2wr2GIoxg2Jpr0LaZPQ02JWI3uMoyubzblmmWXBbGmpBNCOUevHcCys/ACQReKIPyPr" } }, { "type": "step", "result": "=x^{2}-1-2x^{3}+2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p6TdZultUU9+l7D5ThU+pz2gYtQEl4V79ys4pDeUXjWjkVi15I8rBefLi4Iyt2wrJFkNSRU7e5ZFRmsd/DprMTZ5XH2EsTyILKdRR+U00GD8bYA0b6V2RSTOZ7Os9NODWuWICEFO0JF/SuViJ+KhdEVdR1QV4KUbMJKLFci5OIA=" } }, { "type": "interim", "title": "Simplify $$x^{3}+x-x^{2}+x^{2}-1-2x^{3}+2x:{\\quad}-x^{3}+3x-1$$", "input": "x^{3}+x-x^{2}+x^{2}-1-2x^{3}+2x", "result": "=-x^{3}+3x-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=x^{3}-2x^{3}-x^{2}+x^{2}+x+2x-1" }, { "type": "step", "primary": "Add similar elements: $$-x^{2}+x^{2}=0$$", "result": "=x^{3}-2x^{3}+x+2x-1" }, { "type": "step", "primary": "Add similar elements: $$x^{3}-2x^{3}=-x^{3}$$", "result": "=-x^{3}+x+2x-1" }, { "type": "step", "primary": "Add similar elements: $$x+2x=3x$$", "result": "=-x^{3}+3x-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tpBNpyd17bRheINskotouEXqoFdW+S9znedHb52AZVEAlilG71elit3w1IBbYN0PkHkPJHC8driElM+4JZP9LGyhSC0tvvRpcrj4JUJNEzJ6pfF1z6umzUJTJvt+ojYZsMcYC9/IkrIbye/YjuvnYpMzSIPtMHYF0gm1O5wBKWKWL1yMAxT4MNWm83vxFqsD" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=-4\\left(x^{2}-x+1\\right)\\left(-x^{3}+3x-1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}:{\\quad}\\left(x^{2}-x+1\\right)^{4}$$", "input": "\\left(\\left(x^{2}-x+1\\right)^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(x^{2}-x+1\\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}-x+1\\right)^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTK79HpTUoF82HJ6EXuPUH0dPooFi9YZ7zCpVDwEMMsJQJZuTAY5js+oqjdT8ksltuU7Wx0Z22gn0Gx0YcfUOfsicDtr1/4SZLlnwrW0smPuQCM/vqpbrqU5SxRHBPSde+KW0O2HBeCRdRomrKG01Fp/JsVeMbB1fDmYqmludnc=" } }, { "type": "step", "result": "=-\\frac{4\\left(x^{2}-x+1\\right)\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{4}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{2}-x+1$$", "result": "=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tDBnhv4hm4Jgtg9YDh6um2wGx8icDnEUNWpCeHa3493dh/cPFGjeLh37S2cNgMzkieO5NqyOhIvFQ46UW7cz9lF7kzcCCgOPA2sy0CXQtSZU4kXWq0uVmlMI1nL7h2Z4o5FYteSPKwXny4uCMrdsK5xTXxBQDep+sOkI8SVoKbt78YH27IOq3qo330yFZNGx0fE1wSJMLHvYS7pxeDAnE3ql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HiSdvlwsj6ffoukxc7R+t09XAZJsKoJK6ZxUWKh7bb/ZarztdMScTFWYio0Twf+XM+0E66iNYu28S1Bwk8YjYeQcI4Gbs0v1XcZSah1I3MUU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}=0:{\\quad}x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…,\\:x\\approx\\:-1.87938…$$", "input": "-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\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": "4\\left(-x^{3}+3x-1\\right)=0" }, { "type": "interim", "title": "Solve $$4\\left(-x^{3}+3x-1\\right)=0:{\\quad}x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…,\\:x\\approx\\:-1.87938…$$", "input": "4\\left(-x^{3}+3x-1\\right)=0", "steps": [ { "type": "interim", "title": "Expand $$4\\left(-x^{3}+3x-1\\right):{\\quad}-4x^{3}+12x-4$$", "input": "4\\left(-x^{3}+3x-1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=4\\left(-x^{3}\\right)+4\\cdot\\:3x+4\\left(-1\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-4x^{3}+4\\cdot\\:3x-4\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$-4x^{3}+4\\cdot\\:3x-4\\cdot\\:1:{\\quad}-4x^{3}+12x-4$$", "input": "-4x^{3}+4\\cdot\\:3x-4\\cdot\\:1", "result": "=-4x^{3}+12x-4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=-4x^{3}+12x-4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=-4x^{3}+12x-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Riwm73kAYSQivzYjI5vZYPzvBm708WKcKOjF48jNte3oTCAC3hcNW1FUsqCI31nft2g6wAs+7JzewqfesYkp+O9sGZu5A1MXROmEpnxG69pwplAPwRZVqRxWeCJu4KQqv8iaRaxHOpAtfnOgk8icLQ==" } }, { "type": "step", "result": "-4x^{3}+12x-4=0" }, { "type": "interim", "title": "Find one solution for $$-4x^{3}+12x-4=0$$ using Newton-Raphson:$${\\quad}x\\approx\\:0.34729…$$", "input": "-4x^{3}+12x-4=0", "steps": [ { "type": "definition", "title": "Newton-Raphson Approximation Definition", "text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$" }, { "type": "step", "result": "f\\left(x\\right)=-4x^{3}+12x-4" }, { "type": "interim", "title": "Find $$f^{^{\\prime}}\\left(x\\right):{\\quad}-12x^{2}+12$$", "input": "\\frac{d}{dx}\\left(-4x^{3}+12x-4\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(4x^{3}\\right)+\\frac{d}{dx}\\left(12x\\right)-\\frac{d}{dx}\\left(4\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{3}\\right)=12x^{2}$$", "input": "\\frac{d}{dx}\\left(4x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dx}\\left(x^{3}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=12x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuytnrDSBQVECiAN7hMFdqKTdaV09PMxEKZ9FieghTFwbVQDmNnvMzBhKnFOUzUT515NkzKQgtswLlLi9MgL+gq5QV7agSZLIzF7D9vX0CHvx8XaWXbYXWiYPXxbVFoLirCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(12x\\right)=12$$", "input": "\\frac{d}{dx}\\left(12x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=12\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=12", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhF+Ykp5nUgfRlgPAvwoz5KXIQHgliMhSOSNsNni19In94H8CoGnrS97E87MitDaQA4bfwiV6iMLJ5sC1nL7dOZiPtq11pJT4yOnp/GI6P1RsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\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+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=-12x^{2}+12-0" }, { "type": "step", "primary": "Simplify", "result": "=-12x^{2}+12", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Generic Find Title 1Eq" } }, { "type": "step", "primary": "Let $$x_{0}=0$$", "secondary": [ "Compute $$x_{n+1}$$ until $$\\Delta\\:x_{n+1}\\:<\\:0.000001$$" ] }, { "type": "interim", "title": "$$x_{1}=0.33333…{\\quad:\\quad}Δx_{1}=0.33333…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{0}\\right)=-4\\cdot\\:0^{3}+12\\cdot\\:0-4=-4$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{0}\\right)=-12\\cdot\\:0^{2}+12=12$$", "$$x_{1}=0-\\frac{-4}{12}=0.33333…$$" ], "result": "x_{1}=0.33333…" }, { "type": "step", "primary": "$$Δx_{1}=\\left|0.33333…-0\\right|=0.33333…$$", "result": "Δx_{1}=0.33333…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{2}=0.34722…{\\quad:\\quad}Δx_{2}=0.01388…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{1}\\right)=-4\\cdot\\:0.33333…^{3}+12\\cdot\\:0.33333…-4=-0.14814…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{1}\\right)=-12\\cdot\\:0.33333…^{2}+12=10.66666…$$", "$$x_{2}=0.33333…-\\frac{-0.14814…}{10.66666…}=0.34722…$$" ], "result": "x_{2}=0.34722…" }, { "type": "step", "primary": "$$Δx_{2}=\\left|0.34722…-0.33333…\\right|=0.01388…$$", "result": "Δx_{2}=0.01388…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{3}=0.34729…{\\quad:\\quad}Δx_{3}=0.00007…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{2}\\right)=-4\\cdot\\:0.34722…^{3}+12\\cdot\\:0.34722…-4=-0.00078…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{2}\\right)=-12\\cdot\\:0.34722…^{2}+12=10.55324…$$", "$$x_{3}=0.34722…-\\frac{-0.00078…}{10.55324…}=0.34729…$$" ], "result": "x_{3}=0.34729…" }, { "type": "step", "primary": "$$Δx_{3}=\\left|0.34729…-0.34722…\\right|=0.00007…$$", "result": "Δx_{3}=0.00007…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{4}=0.34729…{\\quad:\\quad}Δx_{4}=2.16999E-9$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{3}\\right)=-4\\cdot\\:0.34729…^{3}+12\\cdot\\:0.34729…-4=-2.28991E-8$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{3}\\right)=-12\\cdot\\:0.34729…^{2}+12=10.55262…$$", "$$x_{4}=0.34729…-\\frac{-2.28991E-8}{10.55262…}=0.34729…$$" ], "result": "x_{4}=0.34729…" }, { "type": "step", "primary": "$$Δx_{4}=\\left|0.34729…-0.34729…\\right|=2.16999E-9$$", "result": "Δx_{4}=2.16999E-9" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "x\\approx\\:0.34729…" } ], "meta": { "interimType": "Newton Raphson Find Real Solution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjBsq0XbafKOIsiR0aoui88aI9RwWTsFLx5YFa51QVTzOhisOsUeHnX6OraBu6ek9Mf1+TQKJ10qsZyr5w2pAkWVmiJYZWKJdDF6H91U4YoI52MBy0OGB5sWglSw8yYQ1iq50Cy/b6GkFfiDNL2U+B+N7YlIOlAs02gZJbr3p5jnL+eFUAsSXAw1OEzLYTWSteNkUhf4ZttL/2mBQo1S2aLrQFs9JztE6/D/ToPcRIqqiM0d87BidXtowt6he4Z9cWO/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Apply long division:$${\\quad}\\frac{-4x^{3}+12x-4}{x-0.34729…}=-4x^{2}-1.38918…x+11.51754…$$" }, { "type": "step", "result": "-4x^{2}-1.38918…x+11.51754…\\approx\\:0" }, { "type": "interim", "title": "Find one solution for $$-4x^{2}-1.38918…x+11.51754…=0$$ using Newton-Raphson:$${\\quad}x\\approx\\:1.53208…$$", "input": "-4x^{2}-1.38918…x+11.51754…=0", "steps": [ { "type": "definition", "title": "Newton-Raphson Approximation Definition", "text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$" }, { "type": "step", "result": "f\\left(x\\right)=-4x^{2}-1.38918…x+11.51754…" }, { "type": "interim", "title": "Find $$f^{^{\\prime}}\\left(x\\right):{\\quad}-8x-1.38918…$$", "input": "\\frac{d}{dx}\\left(-4x^{2}-1.38918…x+11.51754…\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(4x^{2}\\right)-\\frac{d}{dx}\\left(1.38918…x\\right)+\\frac{d}{dx}\\left(11.51754…\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{2}\\right)=8x$$", "input": "\\frac{d}{dx}\\left(4x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsGgzDnzN0IPmwhs7frKR8eTdaV09PMxEKZ9FieghTFwbuQyGyH7AL8DepgTwjAnNKN6Hv6MoTMtvtU0IQwXdn/V+9abCUkdC/28BK3pLujtM2vkiNOK25CPGkWDnxgF4w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1.38918…x\\right)=1.38918…$$", "input": "\\frac{d}{dx}\\left(1.38918…x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=1.38918…\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1.38918…\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=1.38918…", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtnZRtKLCXpCmG2yIboSF91s3bzEbqKu8+A9drXdKSFLqKwXnDyHJSOk7SW/uMHpmJbEYfkCJfsT5zotlo6paukOG38IleojCyebAtZy+3TmaotPbZqUVJCpV8qQgTPH0eerXyoFkaPSuiv5rBPCpDmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(11.51754…\\right)=0$$", "input": "\\frac{d}{dx}\\left(11.51754…\\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+idP5vLVrjUll6eMdYr91QuW0++CgKyr3zQx5PqdKSCY1fqKE9KQuJfhrlGtxqKwXnDyHJSOk7SW/uMHpmILlWngNL5BbQEd4M6iPSGdjDT5Dj/fM73/u0bafjbUvGngiKqB48yz5R+PE0NALeCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=-8x-1.38918…+0" }, { "type": "step", "primary": "Simplify", "result": "=-8x-1.38918…", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Generic Find Title 1Eq" } }, { "type": "step", "primary": "Let $$x_{0}=5$$", "secondary": [ "Compute $$x_{n+1}$$ until $$\\Delta\\:x_{n+1}\\:<\\:0.000001$$" ] }, { "type": "interim", "title": "$$x_{1}=2.69436…{\\quad:\\quad}Δx_{1}=2.30563…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{0}\\right)=-4\\cdot\\:5^{2}-1.38918…\\cdot\\:5+11.51754…=-95.42838…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{0}\\right)=-8\\cdot\\:5-1.38918…=-41.38918…$$", "$$x_{1}=5-\\frac{-95.42838…}{-41.38918…}=2.69436…$$" ], "result": "x_{1}=2.69436…" }, { "type": "step", "primary": "$$Δx_{1}=\\left|2.69436…-5\\right|=2.30563…$$", "result": "Δx_{1}=2.30563…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{2}=1.76759…{\\quad:\\quad}Δx_{2}=0.92676…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{1}\\right)=-4\\cdot\\:2.69436…^{2}-1.38918…\\cdot\\:2.69436…+11.51754…=-21.26382…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{1}\\right)=-8\\cdot\\:2.69436…-1.38918…=-22.94409…$$", "$$x_{2}=2.69436…-\\frac{-21.26382…}{-22.94409…}=1.76759…$$" ], "result": "x_{2}=1.76759…" }, { "type": "step", "primary": "$$Δx_{2}=\\left|1.76759…-2.69436…\\right|=0.92676…$$", "result": "Δx_{2}=0.92676…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{3}=1.54637…{\\quad:\\quad}Δx_{3}=0.22122…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{2}\\right)=-4\\cdot\\:1.76759…^{2}-1.38918…\\cdot\\:1.76759…+11.51754…=-3.43558…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{2}\\right)=-8\\cdot\\:1.76759…-1.38918…=-15.52996…$$", "$$x_{3}=1.76759…-\\frac{-3.43558…}{-15.52996…}=1.54637…$$" ], "result": "x_{3}=1.54637…" }, { "type": "step", "primary": "$$Δx_{3}=\\left|1.54637…-1.76759…\\right|=0.22122…$$", "result": "Δx_{3}=0.22122…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{4}=1.53214…{\\quad:\\quad}Δx_{4}=0.01422…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{3}\\right)=-4\\cdot\\:1.54637…^{2}-1.38918…\\cdot\\:1.54637…+11.51754…=-0.19575…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{3}\\right)=-8\\cdot\\:1.54637…-1.38918…=-13.76018…$$", "$$x_{4}=1.54637…-\\frac{-0.19575…}{-13.76018…}=1.53214…$$" ], "result": "x_{4}=1.53214…" }, { "type": "step", "primary": "$$Δx_{4}=\\left|1.53214…-1.54637…\\right|=0.01422…$$", "result": "Δx_{4}=0.01422…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{5}=1.53208…{\\quad:\\quad}Δx_{5}=0.00005…$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{4}\\right)=-4\\cdot\\:1.53214…^{2}-1.38918…\\cdot\\:1.53214…+11.51754…=-0.00080…$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{4}\\right)=-8\\cdot\\:1.53214…-1.38918…=-13.64637…$$", "$$x_{5}=1.53214…-\\frac{-0.00080…}{-13.64637…}=1.53208…$$" ], "result": "x_{5}=1.53208…" }, { "type": "step", "primary": "$$Δx_{5}=\\left|1.53208…-1.53214…\\right|=0.00005…$$", "result": "Δx_{5}=0.00005…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$x_{6}=1.53208…{\\quad:\\quad}Δx_{6}=1.03164E-9$$", "steps": [ { "type": "step", "primary": "$$f\\left(x_{5}\\right)=-4\\cdot\\:1.53208…^{2}-1.38918…\\cdot\\:1.53208…+11.51754…=-1.40777E-8$$", "secondary": [ "$$f^{^{\\prime}}\\left(x_{5}\\right)=-8\\cdot\\:1.53208…-1.38918…=-13.64589…$$", "$$x_{6}=1.53208…-\\frac{-1.40777E-8}{-13.64589…}=1.53208…$$" ], "result": "x_{6}=1.53208…" }, { "type": "step", "primary": "$$Δx_{6}=\\left|1.53208…-1.53208…\\right|=1.03164E-9$$", "result": "Δx_{6}=1.03164E-9" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "x\\approx\\:1.53208…" } ], "meta": { "interimType": "Newton Raphson Find Real Solution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjBsq0XbafKOIsiR0aoui88au7ftEABzhD3ot+La6PO4I7WsvJtbp7P5ddLGfFzHK3p1w4NWdymWGg64uL7J8/NORsKPEs71aM2voPbmwOHY4s+qZS/Zb+4dtNu7SZB+jI5hN8B+9G2wkcVLo0RX+B1iP7qAqF+PAISiydrQpUxLW25KB+jBxoNJrTrINu1WPPYglgMifIilWxOZOKJ2/iQdl6bk6edxTixDgdV003BzPieYN2CTNylEloCRSYAH0sUOmsaD8LMLVaTFz8QwrRyMsjlLglYGfo2yyl2JqommNXMkMik0z8GIqYP+o/57Wi0=" } }, { "type": "step", "primary": "Apply long division:$${\\quad}\\frac{-4x^{2}-1.38918…x+11.51754…}{x-1.53208…}=-4x-7.51754…$$" }, { "type": "step", "result": "-4x-7.51754…\\approx\\:0" }, { "type": "step", "result": "x\\approx\\:-1.87938…" }, { "type": "step", "primary": "The solutions are", "result": "x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…,\\:x\\approx\\:-1.87938…" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…,\\:x\\approx\\:-1.87938…" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-1.87938…,\\:x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x1nUSsx/WhOV8CTUVRVLWnYRaqIBKrsqDbVlgsfHICCrpUYdTe3zE3xR93k5g33LF+7C5h4xtiIO37VrzELRk9buECOGOD/YaR+WO3nMRpgceBZsqElWBiXq7uu1FK2sRvE3Nmuf9f5QX+2UNatIM4moUpK1G+jYScZ6BMfF61iDQGvq49Gr/0VN+XxhDo52zqwRPaARnrEimuefAiYIpQzxBZYtyVsiDh3Qzw0ZPlk7ppVxRlfkz1Zg1KP4pT1RzYKVYMb2hovWQR/ncbG2E6Y=" } }, { "type": "step", "primary": "Identify inflection points not in domain or where $$f\\left(x\\right)$$ is not continuous" }, { "type": "interim", "title": "Domain of $$\\frac{x^{2}+x+1}{x^{2}-x+1}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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": "step", "primary": "All inflection points are in domain and not on domain edges", "result": "x=-1.87938…,\\:x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…" }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Concave Downward$$:-\\infty\\:<x<-1.87938…,\\:$$Concave Upward$$:-1.87938…<x<0.34729…,\\:$$Concave Downward$$:0.34729…<x<1.53208…,\\:$$Concave Upward$$:1.53208…<x<\\infty\\:$$", "steps": [ { "type": "step", "primary": "Combine inflection point(s): $$x=-1.87938…,\\:x\\approx\\:0.34729…,\\:x\\approx\\:1.53208…$$", "secondary": [ "With the domain to get $$f''\\left(x\\right)$$ sign intervals:" ], "result": "-\\infty\\:<x<-1.87938…,\\:-1.87938…<x<0.34729…,\\:0.34729…<x<1.53208…,\\:1.53208…<x<\\infty\\:" }, { "type": "interim", "title": "Check the sign of $$f\\:{^{\\prime\\prime}}\\left(x\\right)=-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$ at each sign interval", "steps": [ { "type": "interim", "title": "Check the sign of $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$ at $$-\\infty\\:<x<-1.87938…:{\\quad}$$Negative", "steps": [ { "type": "step", "primary": "Evaluate $$f''\\left(x\\right)$$ at a point on the interval. Take the point $$x=-3$$ and plug it into $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "result": "-\\frac{4\\left(-\\left(-3\\right)^{3}+3\\left(-3\\right)-1\\right)}{\\left(\\left(-3\\right)^{2}-\\left(-3\\right)+1\\right)^{3}}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{68}{2197}" }, { "type": "step", "result": "\\mathrm{Negative}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "interim", "title": "Check the sign of $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$ at $$-1.87938…<x<0.34729…:{\\quad}$$Positive", "steps": [ { "type": "step", "primary": "Evaluate $$f''\\left(x\\right)$$ at a point on the interval. Take the point $$x=-1$$ and plug it into $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "result": "-\\frac{4\\left(-\\left(-1\\right)^{3}+3\\left(-1\\right)-1\\right)}{\\left(\\left(-1\\right)^{2}-\\left(-1\\right)+1\\right)^{3}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{4}{9}" }, { "type": "step", "result": "\\mathrm{Positive}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "interim", "title": "Check the sign of $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$ at $$0.34729…<x<1.53208…:{\\quad}$$Negative", "steps": [ { "type": "step", "primary": "Evaluate $$f''\\left(x\\right)$$ at a point on the interval. Take the point $$x=1$$ and plug it into $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "result": "-\\frac{4\\left(-1^{3}+3\\cdot\\:1-1\\right)}{\\left(1^{2}-1+1\\right)^{3}}" }, { "type": "step", "primary": "Simplify", "result": "-4" }, { "type": "step", "result": "\\mathrm{Negative}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "interim", "title": "Check the sign of $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$ at $$1.53208…<x<\\infty\\::{\\quad}$$Positive", "steps": [ { "type": "step", "primary": "Evaluate $$f''\\left(x\\right)$$ at a point on the interval. Take the point $$x=3$$ and plug it into $$-\\frac{4\\left(-x^{3}+3x-1\\right)}{\\left(x^{2}-x+1\\right)^{3}}$$", "result": "-\\frac{4\\left(-3^{3}+3\\cdot\\:3-1\\right)}{\\left(3^{2}-3+1\\right)^{3}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{76}{343}" }, { "type": "step", "result": "\\mathrm{Positive}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } } ], "meta": { "interimType": "Check Interval Sign 1Eq" } }, { "type": "step", "primary": "Summary of the sign intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|c|c|}\\hline &-\\infty <x<-1.87938…&x=-1.87938…&-1.87938…<x<0.34729…&x\\approx 0.34729…&0.34729…<x<1.53208…&x\\approx 1.53208…&1.53208…<x<\\infty \\\\\\hline \\mathrm{Sign}&-&0&+&0&-&0&+\\\\\\hline \\mathrm{Behavior}&\\mathrm{Concave\\:Downward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Upward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Downward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Upward}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Concave\\:Downward}:-\\infty\\:<x<-1.87938…,\\:\\mathrm{Concave\\:Upward}:-1.87938…<x<0.34729…,\\:\\mathrm{Concave\\:Downward}:0.34729…<x<1.53208…,\\:\\mathrm{Concave\\:Upward}:1.53208…<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq" } }, { "type": "interim", "title": "Plug $$x=-1.87938…\\:$$into $$\\frac{x^{2}+x+1}{x^{2}-x+1}:{\\quad}0.41374…$$", "input": "\\frac{\\left(-1.87938…\\right)^{2}+\\left(-1.87938…\\right)+1}{\\left(-1.87938…\\right)^{2}-\\left(-1.87938…\\right)+1}", "result": "\\left(-1.87938…,\\:0.41374…\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "0.41374…" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cN+kAkmLVhzJImoLmd9+2VXJln3hYWGvetv33m/Pl1OEwuIzntyFGItIRHwKrXpO1csIT2H8ijzrtr9wclGZ/KeRGE8m0MgwXnydOpOAGZvCLD0BHfIjbCrC8fvdFrOFkZO5XGPQsj9OMCKt1RprChfiQKmgLvFVUqBQYonmk/8TXSbS6nauzDnHrFvpyMvLg" } }, { "type": "interim", "title": "Plug $$x=0.34729…\\:$$into $$\\frac{x^{2}+x+1}{x^{2}-x+1}:{\\quad}1.89819…$$", "input": "\\frac{0.34729…^{2}+0.34729…+1}{0.34729…^{2}-0.34729…+1}", "result": "\\left(0.34729…,\\:1.89819…\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "1.89819…" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7csYYC3TupIR9YB8ehHdz3HtYhxrMCRROByfoecEjwjGPJEf7HlfPo6drgrTRhhBawWFYzcA4oaZR79v7JdxE/FgtdJ+3fdfCz/M30Niaqh0gsV11gcRaJhtCiGQhEkpEhTq+5b05PT9qN4rE2dCRUpkqyMq4SEs/1NUZePmrKiXG6Gr6BipWkrtySfNl1de2q" } }, { "type": "interim", "title": "Plug $$x=1.53208…\\:$$into $$\\frac{x^{2}+x+1}{x^{2}-x+1}:{\\quad}2.68805…$$", "input": "\\frac{1.53208…^{2}+1.53208…+1}{1.53208…^{2}-1.53208…+1}", "result": "\\left(1.53208…,\\:2.68805…\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "2.68805…" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cj6SSs5tu8cKIpARH5Y0hb9YhxrMCRROByfoecEjwjGPJEf7HlfPo6drgrTRhhBawWFYzcA4oaZR79v7JdxE/FgtdJ+3fdfCz/M30Niaqh0hJueg31rE6sEv2JyYtvGp5Tq+5b05PT9qN4rE2dCRUpkqyMq4SEs/1NUZePmrKiXG6Gr6BipWkrtySfNl1de2q" } }, { "type": "step", "result": "\\left(-1.87938…,\\:0.41374…\\right),\\:\\left(0.34729…,\\:1.89819…\\right),\\:\\left(1.53208…,\\:2.68805…\\right)" } ], "meta": { "solvingClass": "Function Inflection" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x^{2}+x+1}{x^{2}-x+1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }