tanh^2(x)+5sech(x)-5=0

{ "query": { "display": "$$\\tanh^{2}\\left(x\\right)+5\\sech\\left(x\\right)-5=0$$", "symbolab_question": "EQUATION#\\tanh^{2}(x)+5\\sech(x)-5=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0", "degrees": "x=0^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tanh^{2}\\left(x\\right)+5\\sech\\left(x\\right)-5=0{\\quad:\\quad}x=0$$", "input": "\\tanh^{2}\\left(x\\right)+5\\sech\\left(x\\right)-5=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\tanh^{2}\\left(x\\right)+5\\sech\\left(x\\right)-5=0", "result": "\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+e^{-x}}-5=0", "steps": [ { "type": "step", "primary": "Use the Hyperbolic identity: $$\\tanh\\left(x\\right)=\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$", "result": "\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\sech\\left(x\\right)-5=0" }, { "type": "step", "primary": "Use the Hyperbolic identity: $$\\sech\\left(x\\right)=\\frac{2}{e^{x}+e^{-x}}$$", "result": "\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+e^{-x}}-5=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U6oLQhfF7xIbIMXYERU996QX2Rz0GvXAtEeA0e6vtIlSaG6HaLO+F18lgErkiu89glS0lOYo+MEX6Bpkgk4kWW2OqI7u8uw/2TDZOKR8MlHIjRu93pF6Ud8rAuy8zIsmQa8BpqbwL52/fLYMlzn6n7g3xJtlUNz/2E5XQK4kQb9VRpjEar7/vLFBnIjppKZ7DVtw76i2lmRqKwCkJeLIcarjr757PdX+g6wdxD40rur/5utQUvpyWuzqGdNN8sTzSYU/rlY9QcIgZOnNP8QBxx62DrT5lSWo/mMH5F6V1QDkAxHFda36329rSpnJqDHJ" } }, { "type": "interim", "title": "$$\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+e^{-x}}-5=0{\\quad:\\quad}x=0$$", "input": "\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+e^{-x}}-5=0", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "\\left(\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+e^{-x}}-5=0", "result": "\\left(\\frac{e^{x}-\\left(e^{x}\\right)^{-1}}{e^{x}+\\left(e^{x}\\right)^{-1}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+\\left(e^{x}\\right)^{-1}}-5=0", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "secondary": [ "$$e^{-x}=\\left(e^{x}\\right)^{-1}$$" ], "result": "\\left(\\frac{e^{x}-\\left(e^{x}\\right)^{-1}}{e^{x}+\\left(e^{x}\\right)^{-1}}\\right)^{2}+5\\cdot\\:\\frac{2}{e^{x}+\\left(e^{x}\\right)^{-1}}-5=0", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PYIowdXt2t1vjFLpvo+g6KDLHMMUZuXEn+d7lqR+uWFT+0sW0yjjIylLRpvNW4pFQRRf0FBfE5tCilRSivzvSMaIQ/wW65x0L6Z1chzSWE6R37vWAGIG3TSz4jIasZmqJjt3NlW+9eZYpv1ldFQ9lvYGXcnzpvtX73vvaMbgjVk/f2lahIyMqfrOREolnrDvCGTDxOfIm+VlpirxV5tKKu3WGEdfvHu14b1WfUNTKSCpCk1Okrr4CJXhvSk8fqU0G4cOcikkLgTARwKJ6kvrEH1Bki9ZsORIzODp2Ta6w02OPs26/xDAcHepyqXvhh2v1sD7NfhsPe7eDHrmjY0mE9IlGhTocLpuLor5cjpkiIgcVBYWZA09tgc0INvET0Hn" } }, { "type": "step", "primary": "Rewrite the equation with $$e^{x}=u$$", "result": "\\left(\\frac{u-\\left(u\\right)^{-1}}{u+\\left(u\\right)^{-1}}\\right)^{2}+5\\cdot\\:\\frac{2}{u+\\left(u\\right)^{-1}}-5=0" }, { "type": "interim", "title": "Solve $$\\left(\\frac{u-u^{-1}}{u+u^{-1}}\\right)^{2}+5\\cdot\\:\\frac{2}{u+u^{-1}}-5=0:{\\quad}u=1$$", "input": "\\left(\\frac{u-u^{-1}}{u+u^{-1}}\\right)^{2}+5\\cdot\\:\\frac{2}{u+u^{-1}}-5=0", "steps": [ { "type": "step", "primary": "Refine", "result": "\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}+\\frac{10u}{u^{2}+1}-5=0" }, { "type": "interim", "title": "Multiply by LCM", "input": "\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}+\\frac{10u}{u^{2}+1}-5=0", "result": "\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}=0", "steps": [ { "type": "interim", "title": "Find Least Common Multiplier of $$\\left(u^{2}+1\\right)^{2},\\:u^{2}+1:{\\quad}\\left(u^{2}+1\\right)^{2}$$", "input": "\\left(u^{2}+1\\right)^{2},\\:u^{2}+1", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$\\left(u^{2}+1\\right)^{2}$$ or $$u^{2}+1$$", "result": "=\\left(u^{2}+1\\right)^{2}" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top in Equation Title 1Eq" } }, { "type": "step", "primary": "Multiply by LCM=$$\\left(u^{2}+1\\right)^{2}$$", "result": "\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}\\left(u^{2}+1\\right)^{2}+\\frac{10u}{u^{2}+1}\\left(u^{2}+1\\right)^{2}-5\\left(u^{2}+1\\right)^{2}=0\\cdot\\:\\left(u^{2}+1\\right)^{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}\\left(u^{2}+1\\right)^{2}+\\frac{10u}{u^{2}+1}\\left(u^{2}+1\\right)^{2}-5\\left(u^{2}+1\\right)^{2}=0\\cdot\\:\\left(u^{2}+1\\right)^{2}", "result": "\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}=0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}\\left(u^{2}+1\\right)^{2}:{\\quad}\\left(u^{2}-1\\right)^{2}$$", "input": "\\frac{\\left(u^{2}-1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(u^{2}-1\\right)^{2}\\left(u^{2}+1\\right)^{2}}{\\left(u^{2}+1\\right)^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\left(u^{2}+1\\right)^{2}$$", "result": "=\\left(u^{2}-1\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73DGSNcEoUP5y+9/WZYagl2+qaFvFfY/83cvDPGOqlBgruFfiOKwmTEQdIrKBO+DPJ1UGfY00IVmqDhmhWHpTCqORWLXkjysF58uLgjK3bCvZcVjFFJGJtKph2/zIr/KtRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rNpfKH+tRjjzu81syH3RXyz1BAr+qrfwcA+CU7kVYagpulggbf7rqmqtfPVCA4X9aJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Simplify $$\\frac{10u}{u^{2}+1}\\left(u^{2}+1\\right)^{2}:{\\quad}10u\\left(u^{2}+1\\right)$$", "input": "\\frac{10u}{u^{2}+1}\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{10u\\left(u^{2}+1\\right)^{2}}{u^{2}+1}" }, { "type": "step", "primary": "Cancel the common factor: $$u^{2}+1$$", "result": "=10u\\left(u^{2}+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74kQ9rgvkQzqE40P++kFHAL7J5MXod2DhPYOMaaym2V8nVQZ9jTQhWaoOGaFYelMKo5FYteSPKwXny4uCMrdsKzYaF+0+zAAXKoH5BLFE/tTvbBmbuQNTF0TphKZ8Ruva8pOMITYik9N8AtIc49Ww1X3L/qXfgo5HGl5+KP5d3BtrctlLONYpGCswSXk6K23esIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$0\\cdot\\:\\left(u^{2}+1\\right)^{2}:{\\quad}0$$", "input": "0\\cdot\\:\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DclvDkRV2DCjR630SN/6QZ9wuFWY2iSOqJSsGXMs9rcJQJZuTAY5js+oqjdT8kslKXPrgUnq5rRq9Cvw1ceDcosaY2cs9YZ/08feWmaOgkg4W37nFKWEkKCD7cUJzr/PkBsewhK5IoUMVwmS7qnwDQ==" } }, { "type": "step", "result": "\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Equation LCM Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAjYcBSbJJKcO5nwp2W+NzB0/DAxjtigTtI+IaWKuzbbSVLm/6YaIi0+7ZvEGaeyY57O+uv4rIwBgO9VTm19InvNhgDk3ar6yeADuqWrtmUS6ara+52Td5xcUnwkQYXwhTNBARJ5th8k9d8tRkH2rJEo3oe/oyhMy2+1TQhDBd2f60ISZzDcJTIRwqEQb3IwO20fcrxFdjUdU7pbuurYJ8Y" } }, { "type": "interim", "title": "Solve $$\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}=0:{\\quad}u=1$$", "input": "\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}=0", "steps": [ { "type": "interim", "title": "Factor $$\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}:{\\quad}-2\\left(u-1\\right)^{2}\\left(2u^{2}-u+2\\right)$$", "input": "\\left(u^{2}-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(u^{2}-1\\right)^{2}=\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}$$", "input": "\\left(u^{2}-1\\right)^{2}", "steps": [ { "type": "interim", "title": "Factor $$\\left(u^{2}-1\\right)^{2}:{\\quad}\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}$$", "steps": [ { "type": "interim", "title": "Factor $$u^{2}-1:{\\quad}\\left(u+1\\right)\\left(u-1\\right)$$", "input": "u^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=u^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$u^{2}-1^{2}=\\left(u+1\\right)\\left(u-1\\right)$$" ], "result": "=\\left(u+1\\right)\\left(u-1\\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(\\left(u+1\\right)\\left(u-1\\right)\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(ab\\right)^n=a^{n}b^{n}$$", "result": "=\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Cn8Q3x1ww5DxfFlJ9gM6AC061ljBSPJeENOw2efoSWsmm7ZqFFiWERn+XApifvR194xavmpU9AOGRbQV99wZv0Ln7e9BNpm+B7gB3+PxPi9im6PyrlQHElNHPVAIzxVTwtlVR/+ACiCB+p1H2hrFkyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}:{\\quad}-4u^{4}+10u^{3}-12u^{2}+10u-4$$", "input": "\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}+10u\\left(u^{2}+1\\right)-5\\left(u^{2}+1\\right)^{2}", "result": "=-4u^{4}+10u^{3}-12u^{2}+10u-4", "steps": [ { "type": "interim", "title": "$$\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}=\\left(u^{2}+2u+1\\right)\\left(u^{2}-2u+1\\right)$$", "input": "\\left(u+1\\right)^{2}\\left(u-1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(u+1\\right)^{2}=u^{2}+2u+1$$", "input": "\\left(u+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=u,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=u^{2}+2u\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$u^{2}+2u\\cdot\\:1+1^{2}:{\\quad}u^{2}+2u+1$$", "input": "u^{2}+2u\\cdot\\:1+1^{2}", "result": "=u^{2}+2u+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=u^{2}+2\\cdot\\:1\\cdot\\:u+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=u^{2}+2u+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D5jFkpO9mAOZ2a8VW90CN913jtrSFDx+UNsawjlOjV0dCcIsixV/POP/HKQDsADtiyUxM3aw1a2ii/L6h9E/N1BChAp8+JH1g/pisZ/gdD4JaeJYtOCzPB7zpugaasP1" } }, { "type": "step", "result": "=\\left(u^{2}+2u+1\\right)\\left(u-1\\right)^{2}" }, { "type": "interim", "title": "$$\\left(u-1\\right)^{2}=u^{2}-2u+1$$", "input": "\\left(u-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=u,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=u^{2}-2u\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$u^{2}-2u\\cdot\\:1+1^{2}:{\\quad}u^{2}-2u+1$$", "input": "u^{2}-2u\\cdot\\:1+1^{2}", "result": "=u^{2}-2u+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=u^{2}-2\\cdot\\:1\\cdot\\:u+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=u^{2}-2u+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLmtYIsTkXbnPh+psoakn913jtrSFDx+UNsawjlOjV1fFS56G5ain6hE8r+WYPyTiyUxM3aw1a2ii/L6h9E/Nx8VHcja8U5Afp2YGYgAHitZtWS5NXL0MwNfYpd5DY7C" } }, { "type": "step", "result": "=\\left(u^{2}+2u+1\\right)\\left(u^{2}-2u+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Mp9dytuFJ42KBmEIH+GUYo5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdj4EL5EV6OaOBnWq3a5Hsx19uL0nzPYvhcZst3gbAySoA2wXHvdwyIpL2rF/iJLCSYpoMZQoTmSoi5XRmE0a0osvPQ5mlbXc4lcCo9UvVjGon1YRBZBW3O37v2gapoJlr2" } }, { "type": "interim", "title": "$$5\\left(u^{2}+1\\right)^{2}=5\\left(u^{4}+2u^{2}+1\\right)$$", "input": "5\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(u^{2}+1\\right)^{2}=u^{4}+2u^{2}+1$$", "input": "\\left(u^{2}+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=u^{2},\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=\\left(u^{2}\\right)^{2}+2u^{2}\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$\\left(u^{2}\\right)^{2}+2u^{2}\\cdot\\:1+1^{2}:{\\quad}u^{4}+2u^{2}+1$$", "input": "\\left(u^{2}\\right)^{2}+2u^{2}\\cdot\\:1+1^{2}", "result": "=u^{4}+2u^{2}+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(u^{2}\\right)^{2}+2\\cdot\\:1\\cdot\\:u^{2}+1" }, { "type": "interim", "title": "$$\\left(u^{2}\\right)^{2}=u^{4}$$", "input": "\\left(u^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=u^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=u^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tfQEFTRzwrlNY77fWE8eXt6GQqufR6tr2vPxOUv7H+90lBTGQrQXJVGyx4QZ+nyA8SrqrDW4mFcEK+hPNqZN8q5zBT3YkWaFAT7u2hcNCpmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$2u^{2}\\cdot\\:1=2u^{2}$$", "input": "2u^{2}\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s7iDY2IE4UnYV6Fmrgg+ngCWKUbvV6WK3fDUgFtg3Q+ixjF5rcLg4t65bWEjZTMGXk2TMpCC2zAuUuL0yAv6Csg9mZIqvOpjJqA8V4ZZCBNu37pDjlCbVm9fwNHZbvFA" } }, { "type": "step", "result": "=u^{4}+2u^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LOiB+HiQjIFanGzIuCXr2y061ljBSPJeENOw2efoSWsDFioBOZrzR54zFY57M1rsf7uSX6F5v9v81DHmITwvfhUodnWZy01nimYzMWi2wZ9ZK6movGuN20eQKVdl8YzGF8EAgfsiqw39HjxsDxlpkA==" } }, { "type": "step", "result": "=5\\left(u^{4}+2u^{2}+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DsedghEBTYS6TLX2dD/DRQCWKUbvV6WK3fDUgFtg3Q8TWT4U7m5ckTFa07+m/FFc3j5KoADXFpd7+mSBJ2iX1INtPy5r2a8A8BecB5kktBshClITBYqO7KJu4Af2paBqzWkNN0gbmOUr+97DN/UP5iS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(u^{2}+2u+1\\right)\\left(u^{2}-2u+1\\right)+10u\\left(u^{2}+1\\right)-5\\left(u^{4}+2u^{2}+1\\right)" }, { "type": "interim", "title": "Expand $$\\left(u^{2}+2u+1\\right)\\left(u^{2}-2u+1\\right):{\\quad}u^{4}-2u^{2}+1$$", "input": "\\left(u^{2}+2u+1\\right)\\left(u^{2}-2u+1\\right)", "result": "=u^{4}-2u^{2}+1+10u\\left(u^{2}+1\\right)-5\\left(u^{4}+2u^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=u^{2}u^{2}+u^{2}\\left(-2u\\right)+u^{2}\\cdot\\:1+2uu^{2}+2u\\left(-2u\\right)+2u\\cdot\\:1+1\\cdot\\:u^{2}+1\\cdot\\:\\left(-2u\\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": "=u^{2}u^{2}-2u^{2}u+1\\cdot\\:u^{2}+2u^{2}u-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u+1\\cdot\\:u^{2}-1\\cdot\\:2u+1\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$u^{2}u^{2}-2u^{2}u+1\\cdot\\:u^{2}+2u^{2}u-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u+1\\cdot\\:u^{2}-1\\cdot\\:2u+1\\cdot\\:1:{\\quad}u^{4}-2u^{2}+1$$", "input": "u^{2}u^{2}-2u^{2}u+1\\cdot\\:u^{2}+2u^{2}u-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u+1\\cdot\\:u^{2}-1\\cdot\\:2u+1\\cdot\\:1", "result": "=u^{4}-2u^{2}+1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=u^{2}u^{2}-2u^{2}u+1\\cdot\\:u^{2}+2u^{2}u+1\\cdot\\:u^{2}-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u-1\\cdot\\:2u+1\\cdot\\:1" }, { "type": "step", "primary": "Add similar elements: $$1\\cdot\\:u^{2}+1\\cdot\\:u^{2}=2u^{2}$$", "result": "=u^{2}u^{2}-2u^{2}u+2u^{2}+2u^{2}u-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u-1\\cdot\\:2u+1\\cdot\\:1" }, { "type": "step", "primary": "Add similar elements: $$-2u^{2}u+2u^{2}u=0$$", "result": "=u^{2}u^{2}+2u^{2}-2\\cdot\\:2uu+2\\cdot\\:1\\cdot\\:u-1\\cdot\\:2u+1\\cdot\\:1" }, { "type": "step", "primary": "Add similar elements: $$2\\cdot\\:1\\cdot\\:u-1\\cdot\\:2u=0$$", "result": "=u^{2}u^{2}+2u^{2}-2\\cdot\\:2uu+1\\cdot\\:1" }, { "type": "interim", "title": "$$u^{2}u^{2}=u^{4}$$", "input": "u^{2}u^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$u^{2}u^{2}=\\:u^{2+2}$$" ], "result": "=u^{2+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=u^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s768LRGQRyhpLQLhhJPXu3MyAn9lkDfZkicUGkO3EF+IpZ5l3AzwC+nENcRX1Q0pb9YuA6m0o+8K8U8s4gDnAXjL2DGRG+U3rAOzULn3KKXEgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\cdot\\:2uu=4u^{2}$$", "input": "2\\cdot\\:2uu", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4uu" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=4u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DJIBMvelQA6NKufR3p1YZ96GQqufR6tr2vPxOUv7H++akSE8zt71NPXBtRyCSNgOP8vQyhiD4JSfqjIvcQ7tijR40msVFBK2U3qE+wAIsFKPATzsdMJbIBTfKlGugMwN" } }, { "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": "=u^{4}+2u^{2}-4u^{2}+1" }, { "type": "step", "primary": "Add similar elements: $$2u^{2}-4u^{2}=-2u^{2}$$", "result": "=u^{4}-2u^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CUM0Po1A02mSlqnv8J16wJCkX9VOVh/N1VgPSFPoWq3NGoPE9TME3q+OPmgkv2RQPJJQ3yImXqgy4T9C/vIkzTIpflYTu+fgSQyCYxWO6VUScn/M2sLYBGgxClqzCxM2MGHyMJFlc3C/fo0ttvwxlir2Gg7b63LOthLctOZgLyskt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Expand $$10u\\left(u^{2}+1\\right):{\\quad}10u^{3}+10u$$", "input": "10u\\left(u^{2}+1\\right)", "result": "=u^{4}-2u^{2}+1+10u^{3}+10u-5\\left(u^{4}+2u^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=10u,\\:b=u^{2},\\:c=1$$" ], "result": "=10uu^{2}+10u\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=10u^{2}u+10\\cdot\\:1\\cdot\\:u" }, { "type": "interim", "title": "Simplify $$10u^{2}u+10\\cdot\\:1\\cdot\\:u:{\\quad}10u^{3}+10u$$", "input": "10u^{2}u+10\\cdot\\:1\\cdot\\:u", "result": "=10u^{3}+10u", "steps": [ { "type": "interim", "title": "$$10u^{2}u=10u^{3}$$", "input": "10u^{2}u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$u^{2}u=\\:u^{2+1}$$" ], "result": "=10u^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=10u^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qWceyFYXndjGvM6GvLPpP80ag8T1MwTer44+aCS/ZFD01wg/iFRHhxL0kuyGhxK84pjMMjej8NVIv5HiNeoS5LSoj4+KNq7K95UtOcGVmT0kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$10\\cdot\\:1\\cdot\\:u=10u$$", "input": "10\\cdot\\:1\\cdot\\:u", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$10\\cdot\\:1=10$$", "result": "=10u" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RUZ99f9WTrrDLNArDR8aNea46OWj7eBhOShmT/t95qvMwViaLUXkeD+JukROhWdjjSGbGoEOslfUgNrGeFz2gupbb6FgLNVq7/p3Wz4lKmWntm84jVw3Hrvx4+GrQrbf" } }, { "type": "step", "result": "=10u^{3}+10u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+gRlZq+cO8Do3u7rE9V32VXTSum/z5kLpMzXS1UJIezdi/mKUYI7ABVrj8tdVxebFbb2HZ+oE8TerD8h1Ff7zEDLjX7DcYtB54q3geRNejEoB3lSSSqTsa4udfXtdJoaRV1HVBXgpRswkosVyLk4gA==" } }, { "type": "interim", "title": "Expand $$-5\\left(u^{4}+2u^{2}+1\\right):{\\quad}-5u^{4}-10u^{2}-5$$", "input": "-5\\left(u^{4}+2u^{2}+1\\right)", "result": "=u^{4}-2u^{2}+1+10u^{3}+10u-5u^{4}-10u^{2}-5", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\left(-5\\right)u^{4}+\\left(-5\\right)\\cdot\\:2u^{2}+\\left(-5\\right)\\cdot\\:1", "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": "=-5u^{4}-5\\cdot\\:2u^{2}-5\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$-5u^{4}-5\\cdot\\:2u^{2}-5\\cdot\\:1:{\\quad}-5u^{4}-10u^{2}-5$$", "input": "-5u^{4}-5\\cdot\\:2u^{2}-5\\cdot\\:1", "result": "=-5u^{4}-10u^{2}-5", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:2=10$$", "result": "=-5u^{4}-10u^{2}-5\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:1=5$$", "result": "=-5u^{4}-10u^{2}-5" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e1xLBWXpNsTrZ5W3l14wk1UYzfSy7t7EMjXpWQE38QDMwViaLUXkeD+JukROhWdjaLuMkOfUaGM5VTFA8a+rKYTWpfVoMmKra60eaA0vpZOCcvL2jSzw1VlRcGX4ka6qzgyK143sl5zy81tOc314QQPUYvqD4ThFqPySWXrzVhs=" } }, { "type": "interim", "title": "Simplify $$u^{4}-2u^{2}+1+10u^{3}+10u-5u^{4}-10u^{2}-5:{\\quad}-4u^{4}+10u^{3}-12u^{2}+10u-4$$", "input": "u^{4}-2u^{2}+1+10u^{3}+10u-5u^{4}-10u^{2}-5", "result": "=-4u^{4}+10u^{3}-12u^{2}+10u-4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=u^{4}-5u^{4}+10u^{3}-2u^{2}-10u^{2}+10u+1-5" }, { "type": "step", "primary": "Add similar elements: $$-2u^{2}-10u^{2}=-12u^{2}$$", "result": "=u^{4}-5u^{4}+10u^{3}-12u^{2}+10u+1-5" }, { "type": "step", "primary": "Add similar elements: $$u^{4}-5u^{4}=-4u^{4}$$", "result": "=-4u^{4}+10u^{3}-12u^{2}+10u+1-5" }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-5=-4$$", "result": "=-4u^{4}+10u^{3}-12u^{2}+10u-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Mp9dytuFJ42KBmEIH+GUYmf2crAIDAi9jxLV0GCt69GKZthOWbRJnpn5JV/kDLqdAJYpRu9XpYrd8NSAW2DdD1uoywY5rC8yEvyaVdL1ezfO9/4BhmRF70m0q08jYtEBF/V+lrqCzSmWdtIjyg4yN/C30sSftAIFS6Qkpy19Ikr9Lh8MdLuuLsFJu8B6VOnImmfmOy4M4FtqXuWlKvDs7jmbVuXPTj59evNGt1Hf08CQ+aHKHSIDdeFlQCTCktom" } }, { "type": "interim", "title": "Factor $$-4u^{4}+10u^{3}-12u^{2}+10u-4:{\\quad}-2\\left(u-1\\right)^{2}\\left(2u^{2}-u+2\\right)$$", "input": "-4u^{4}+10u^{3}-12u^{2}+10u-4", "steps": [ { "type": "interim", "title": "Factor out common term $$-2:{\\quad}-2\\left(2u^{4}-5u^{3}+6u^{2}-5u+2\\right)$$", "input": "-4u^{4}+10u^{3}-12u^{2}+10u-4", "steps": [ { "type": "step", "primary": "Rewrite $$4$$ as $$2\\cdot\\:2$$", "secondary": [ "Rewrite $$10$$ as $$2\\cdot\\:5$$", "Rewrite $$12$$ as $$2\\cdot\\:6$$", "Rewrite $$10$$ as $$2\\cdot\\:5$$", "Rewrite $$4$$ as $$2\\cdot\\:2$$" ], "result": "=-2\\cdot\\:2u^{2\\cdot\\:2}+2\\cdot\\:5u^{3}-2\\cdot\\:6u^{2}+2\\cdot\\:5u-2\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-2$$", "result": "=-2\\left(2u^{4}-5u^{3}+6u^{2}-5u+2\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=-2\\left(2u^{4}-5u^{3}+6u^{2}-5u+2\\right)" }, { "type": "interim", "title": "Factor $$2u^{4}-5u^{3}+6u^{2}-5u+2:{\\quad}\\left(u-1\\right)\\left(u-1\\right)\\left(2u^{2}-u+2\\right)$$", "input": "2u^{4}-5u^{3}+6u^{2}-5u+2", "steps": [ { "type": "step", "primary": "Use the rational root theorem", "secondary": [ "$$a_{0}=2,\\:{\\quad}a_{n}=2$$<br/>The dividers of $$a_{0}:{\\quad}1,\\:2,\\:{\\quad}$$The dividers of $$a_{n}:{\\quad}1,\\:2$$<br/>Therefore, check the following rational numbers:$${\\quad}\\pm\\:\\frac{1,\\:2}{1,\\:2}$$<br/>$$\\frac{1}{1}$$ is a root of the expression, so factor out $$u-1$$" ], "result": "=\\left(u-1\\right)\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}", "meta": { "title": { "extension": "For a polynomial equation with integer coefficients:$${\\quad}a_{n}x^{n}+a_{n-1}x^{n-1}+\\ldots+a_{0}$$<br/>If $$a_{0}$$ and $$a_{n}$$ are integers, then if there is a rational solution<br/>it could be found by checking all the numbers produced for $$\\frac{\\pm\\:\\mathrm{dividers\\:of}\\:a_{0}}{\\mathrm{dividers\\:of}\\:a_{n}}$$" } } }, { "type": "interim", "title": "$$\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}=2u^{3}-3u^{2}+3u-2$$", "input": "\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}", "result": "=2u^{3}-3u^{2}+3u-2", "steps": [ { "type": "interim", "title": "Divide $$\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}:{\\quad}\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}=2u^{3}+\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}$$", "result": "=2u^{3}+\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$2u^{4}-5u^{3}+6u^{2}-5u+2$$<br/>and the divisor $$u-1\\::\\:\\frac{2u^{4}}{u}=2u^{3}$$", "result": "\\mathrm{Quotient}=2u^{3}" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$2u^{3}:\\:2u^{4}-2u^{3}$$", "secondary": [ "Subtract $$2u^{4}-2u^{3}$$ from $$2u^{4}-5u^{3}+6u^{2}-5u+2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-3u^{3}+6u^{2}-5u+2" }, { "type": "step", "primary": "Therefore", "result": "\\frac{2u^{4}-5u^{3}+6u^{2}-5u+2}{u-1}=2u^{3}+\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}:{\\quad}\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}=-3u^{2}+\\frac{3u^{2}-5u+2}{u-1}$$", "result": "=2u^{3}-3u^{2}+\\frac{3u^{2}-5u+2}{u-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-3u^{3}+6u^{2}-5u+2$$<br/>and the divisor $$u-1\\::\\:\\frac{-3u^{3}}{u}=-3u^{2}$$", "result": "\\mathrm{Quotient}=-3u^{2}" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$-3u^{2}:\\:-3u^{3}+3u^{2}$$", "secondary": [ "Subtract $$-3u^{3}+3u^{2}$$ from $$-3u^{3}+6u^{2}-5u+2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=3u^{2}-5u+2" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-3u^{3}+6u^{2}-5u+2}{u-1}=-3u^{2}+\\frac{3u^{2}-5u+2}{u-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{3u^{2}-5u+2}{u-1}:{\\quad}\\frac{3u^{2}-5u+2}{u-1}=3u+\\frac{-2u+2}{u-1}$$", "result": "=2u^{3}-3u^{2}+3u+\\frac{-2u+2}{u-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$3u^{2}-5u+2$$<br/>and the divisor $$u-1\\::\\:\\frac{3u^{2}}{u}=3u$$", "result": "\\mathrm{Quotient}=3u" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$3u:\\:3u^{2}-3u$$", "secondary": [ "Subtract $$3u^{2}-3u$$ from $$3u^{2}-5u+2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-2u+2" }, { "type": "step", "primary": "Therefore", "result": "\\frac{3u^{2}-5u+2}{u-1}=3u+\\frac{-2u+2}{u-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-2u+2}{u-1}:{\\quad}\\frac{-2u+2}{u-1}=-2$$", "result": "=2u^{3}-3u^{2}+3u-2", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-2u+2$$<br/>and the divisor $$u-1\\::\\:\\frac{-2u}{u}=-2$$", "result": "\\mathrm{Quotient}=-2" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$-2:\\:-2u+2$$", "secondary": [ "Subtract $$-2u+2$$ from $$-2u+2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=0" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-2u+2}{u-1}=-2" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Factor $$2u^{3}-3u^{2}+3u-2:{\\quad}\\left(u-1\\right)\\left(2u^{2}-u+2\\right)$$", "input": "2u^{3}-3u^{2}+3u-2", "steps": [ { "type": "step", "primary": "Use the rational root theorem", "secondary": [ "$$a_{0}=2,\\:{\\quad}a_{n}=2$$<br/>The dividers of $$a_{0}:{\\quad}1,\\:2,\\:{\\quad}$$The dividers of $$a_{n}:{\\quad}1,\\:2$$<br/>Therefore, check the following rational numbers:$${\\quad}\\pm\\:\\frac{1,\\:2}{1,\\:2}$$<br/>$$\\frac{1}{1}$$ is a root of the expression, so factor out $$u-1$$" ], "result": "=\\left(u-1\\right)\\frac{2u^{3}-3u^{2}+3u-2}{u-1}", "meta": { "title": { "extension": "For a polynomial equation with integer coefficients:$${\\quad}a_{n}x^{n}+a_{n-1}x^{n-1}+\\ldots+a_{0}$$<br/>If $$a_{0}$$ and $$a_{n}$$ are integers, then if there is a rational solution<br/>it could be found by checking all the numbers produced for $$\\frac{\\pm\\:\\mathrm{dividers\\:of}\\:a_{0}}{\\mathrm{dividers\\:of}\\:a_{n}}$$" } } }, { "type": "interim", "title": "$$\\frac{2u^{3}-3u^{2}+3u-2}{u-1}=2u^{2}-u+2$$", "input": "\\frac{2u^{3}-3u^{2}+3u-2}{u-1}", "result": "=2u^{2}-u+2", "steps": [ { "type": "interim", "title": "Divide $$\\frac{2u^{3}-3u^{2}+3u-2}{u-1}:{\\quad}\\frac{2u^{3}-3u^{2}+3u-2}{u-1}=2u^{2}+\\frac{-u^{2}+3u-2}{u-1}$$", "result": "=2u^{2}+\\frac{-u^{2}+3u-2}{u-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$2u^{3}-3u^{2}+3u-2$$<br/>and the divisor $$u-1\\::\\:\\frac{2u^{3}}{u}=2u^{2}$$", "result": "\\mathrm{Quotient}=2u^{2}" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$2u^{2}:\\:2u^{3}-2u^{2}$$", "secondary": [ "Subtract $$2u^{3}-2u^{2}$$ from $$2u^{3}-3u^{2}+3u-2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-u^{2}+3u-2" }, { "type": "step", "primary": "Therefore", "result": "\\frac{2u^{3}-3u^{2}+3u-2}{u-1}=2u^{2}+\\frac{-u^{2}+3u-2}{u-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-u^{2}+3u-2}{u-1}:{\\quad}\\frac{-u^{2}+3u-2}{u-1}=-u+\\frac{2u-2}{u-1}$$", "result": "=2u^{2}-u+\\frac{2u-2}{u-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-u^{2}+3u-2$$<br/>and the divisor $$u-1\\::\\:\\frac{-u^{2}}{u}=-u$$", "result": "\\mathrm{Quotient}=-u" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$-u:\\:-u^{2}+u$$", "secondary": [ "Subtract $$-u^{2}+u$$ from $$-u^{2}+3u-2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=2u-2" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-u^{2}+3u-2}{u-1}=-u+\\frac{2u-2}{u-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{2u-2}{u-1}:{\\quad}\\frac{2u-2}{u-1}=2$$", "result": "=2u^{2}-u+2", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$2u-2$$<br/>and the divisor $$u-1\\::\\:\\frac{2u}{u}=2$$", "result": "\\mathrm{Quotient}=2" }, { "type": "step", "primary": "Multiply $$u-1$$ by $$2:\\:2u-2$$", "secondary": [ "Subtract $$2u-2$$ from $$2u-2$$ to get new remainder" ], "result": "\\mathrm{Remainder}=0" }, { "type": "step", "primary": "Therefore", "result": "\\frac{2u-2}{u-1}=2" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\left(u-1\\right)\\left(2u^{2}-u+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(u-1\\right)\\left(u-1\\right)\\left(2u^{2}-u+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-2\\left(u-1\\right)\\left(u-1\\right)\\left(2u^{2}-u+2\\right)" }, { "type": "step", "primary": "Refine", "result": "=-2\\left(u-1\\right)^{2}\\left(2u^{2}-u+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-2\\left(u-1\\right)^{2}\\left(2u^{2}-u+2\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "-2\\left(u-1\\right)^{2}\\left(2u^{2}-u+2\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "u-1=0\\lor\\:2u^{2}-u+2=0" }, { "type": "interim", "title": "Solve $$u-1=0:{\\quad}u=1$$", "input": "u-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "u-1=0", "result": "u=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "u-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "u=1" } ], "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 $$2u^{2}-u+2=0:{\\quad}$$No Solution for $$u\\in\\mathbb{R}$$", "input": "2u^{2}-u+2=0", "steps": [ { "type": "interim", "title": "Discriminant $$2u^{2}-u+2=0:{\\quad}-15$$", "input": "2u^{2}-u+2=0", "steps": [ { "type": "step", "primary": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the discriminant is $$b^2-4ac$$", "secondary": [ "For $${\\quad}a=2,\\:b=-1,\\:c=2{:\\quad}\\left(-1\\right)^{2}-4\\cdot\\:2\\cdot\\:2$$" ], "result": "\\left(-1\\right)^{2}-4\\cdot\\:2\\cdot\\:2" }, { "type": "interim", "title": "Expand $$\\left(-1\\right)^{2}-4\\cdot\\:2\\cdot\\:2:{\\quad}-15$$", "input": "\\left(-1\\right)^{2}-4\\cdot\\:2\\cdot\\:2", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$4\\cdot\\:2\\cdot\\:2=16$$", "input": "4\\cdot\\:2\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:2=16$$", "result": "=16" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BrBjmPYwjyi388939n5bKbX/WDhfsCbPMG2GGffA2UajkVi15I8rBefLi4Iyt2wrGZdylS5eUeoZdufIX9MJOA0IERTknIByea4Ue9OUx2omiWJ3WoBdB/ZOWa4wNa7h" } }, { "type": "step", "result": "=1-16" }, { "type": "step", "primary": "Subtract the numbers: $$1-16=-15$$", "result": "=-15" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73mtaCpcXG7PluM4wSgc2v4P4GewAjYD8eoNIZYnR8uXF793Q2VmlBK/MLnQmita0dQxShwOg3W+URNAMu5sPa6VYwrogLl29RT6HYd2NJ316pfF1z6umzUJTJvt+ojYZ0icXJKJKG4FDeyd9d0kVjTQqsHVe47/y2heJhgcMHnTAQN9syxBh2csLq0jtXwTb" } }, { "type": "step", "result": "-15" } ], "meta": { "interimType": "Discriminant Title 1Eq" } }, { "type": "step", "primary": "Discriminant cannot be negative for $$u\\in\\mathbb{R}$$" }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:u\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=1" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}u=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\left(\\frac{u-u^{-1}}{u+u^{-1}}\\right)^{2}+5\\frac{2}{u+u^{-1}}-5$$ and compare to zero" }, { "type": "step", "result": "u=0" }, { "type": "step", "primary": "The following points are undefined", "result": "u=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=1" }, { "type": "step", "primary": "Substitute back $$u=e^{x},\\:$$solve for $$x$$" }, { "type": "interim", "title": "Solve $$e^{x}=1:{\\quad}x=0$$", "input": "e^{x}=1", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "e^{x}=1", "result": "x=0", "steps": [ { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(e^{x}\\right)=\\ln\\left(1\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(e^a\\right)=a$$", "secondary": [ "$$\\ln\\left(e^{x}\\right)=x$$" ], "result": "x=\\ln\\left(1\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$\\ln\\left(1\\right):{\\quad}0$$", "input": "\\ln\\left(1\\right)", "steps": [ { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(1\\right)=0$$", "result": "=0", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cJO4Mw9PX2rs+FfXhju0IwOfOVs9mPIqDLV5QIWwt3m4DS9snDRdGFIEJoiNCqQWTeQKHeh69S6dnv9vSoUoFAJyfalcNsA5/wfkdc07YubiAEmXhYw7WsDRrfT9tRiW" } }, { "type": "step", "result": "x=0" } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GC09ZGLlHa7X367X5UU/GW9f69mFY4KpYZIl0v+ArtOhHeJ1xBi321LjY4vsXC9em3FAiPzxVy0umodhDNEdMicbonpXf+B3YOV7JAsBXDzWwPs1+Gw97t4MeuaNjSYT0iUaFOhwum4uivlyOmSIiBxUFhZkDT22BzQg28RPQec=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\tanh^{2}(x)+5\\sech(x)-5" }, "showViewLarger": true } }, "meta": { "showVerify": true } }