f(x)=(2x+1)/(x^2+2)

{ "query": { "display": "$$f\\left(x\\right)=\\frac{2x+1}{x^{2}+2}$$", "symbolab_question": "FUNCTION#f(x)=\\frac{2x+1}{x^{2}+2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{Domain}: -\\infty <x<\\infty <br/>\\mathrm{Range}: -\\frac{1}{2}\\le f(x)\\le 1<br/>\\mathrm{X\\:Intercepts}: (-\\frac{1}{2},0),\\mathrm{Y\\:Intercepts}: (0,\\frac{1}{2})<br/>\\mathrm{Asymptotes}: \\mathrm{Horizontal}\\:y=0<br/>\\mathrm{Extreme\\:Points}: \\mathrm{Minimum}(-2,-\\frac{1}{2}),\\mathrm{Maximum}(1,1)", "interval": "\\mathrm{Domain}: (-\\infty ,\\infty )<br/>\\mathrm{Range}: [-\\frac{1}{2},1]" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Domain of $$\\frac{2x+1}{x^{2}+2}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$\\frac{2x+1}{x^{2}+2}:{\\quad}-\\frac{1}{2}\\le\\:f\\left(x\\right)\\le\\:1$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "Rewrite as", "result": "\\frac{2x+1}{x^{2}+2}=y" }, { "type": "interim", "title": "Multiply both sides by $$x^{2}+2$$", "input": "\\frac{2x+1}{x^{2}+2}=y", "result": "2x+1=y\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Multiply both sides by $$x^{2}+2$$", "result": "\\frac{2x+1}{x^{2}+2}\\left(x^{2}+2\\right)=y\\left(x^{2}+2\\right)" }, { "type": "step", "primary": "Simplify", "result": "2x+1=y\\left(x^{2}+2\\right)" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "The range is the set of y for which the discriminant is greater or equal to zero" }, { "type": "interim", "title": "Discriminant $$2x+1=y\\left(x^{2}+2\\right):{\\quad}4-8y^{2}+4y$$", "input": "2x+1=y\\left(x^{2}+2\\right)", "steps": [ { "type": "interim", "title": "Expand $$y\\left(x^{2}+2\\right):{\\quad}x^{2}y+2y$$", "input": "y\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=y,\\:b=x^{2},\\:c=2$$" ], "result": "=yx^{2}+y\\cdot\\:2", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=x^{2}y+2y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oFGULuTnhGuXD+280CbsKTN2s7Tu4QLkRRj00bh3VC51DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhm3nQkKXLIWo//rBZ9xhEajl+uf6AzAndAjz3Vql+ewDQ==" } }, { "type": "step", "result": "2x+1=x^{2}y+2y" }, { "type": "step", "primary": "Switch sides", "result": "x^{2}y+2y=2x+1" }, { "type": "interim", "title": "Move $$1\\:$$to the left side", "input": "x^{2}y+2y=2x+1", "result": "x^{2}y+2y-1=2x", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}y+2y-1=2x+1-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}y+2y-1=2x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$2x\\:$$to the left side", "input": "x^{2}y+2y-1=2x", "result": "x^{2}y+2y-1-2x=0", "steps": [ { "type": "step", "primary": "Subtract $$2x$$ from both sides", "result": "x^{2}y+2y-1-2x=2x-2x" }, { "type": "step", "primary": "Simplify", "result": "x^{2}y+2y-1-2x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "yx^{2}-2x+2y-1=0" }, { "type": "step", "primary": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the discriminant is $$b^2-4ac$$", "secondary": [ "For $${\\quad}a=y,\\:b=-2,\\:c=2y-1{:\\quad}\\left(-2\\right)^{2}-4y\\left(2y-1\\right)$$" ], "result": "\\left(-2\\right)^{2}-4y\\left(2y-1\\right)" }, { "type": "interim", "title": "Expand $$\\left(-2\\right)^{2}-4y\\left(2y-1\\right):{\\quad}4-8y^{2}+4y$$", "input": "\\left(-2\\right)^{2}-4y\\left(2y-1\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}-4y\\left(2y-1\\right)" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4-4y\\left(2y-1\\right)" }, { "type": "interim", "title": "Expand $$-4y\\left(2y-1\\right):{\\quad}-8y^{2}+4y$$", "input": "-4y\\left(2y-1\\right)", "result": "=4-8y^{2}+4y", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-4y,\\:b=2y,\\:c=1$$" ], "result": "=-4y\\cdot\\:2y-\\left(-4y\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-4\\cdot\\:2yy+4\\cdot\\:1\\cdot\\:y" }, { "type": "interim", "title": "Simplify $$-4\\cdot\\:2yy+4\\cdot\\:1\\cdot\\:y:{\\quad}-8y^{2}+4y$$", "input": "-4\\cdot\\:2yy+4\\cdot\\:1\\cdot\\:y", "result": "=-8y^{2}+4y", "steps": [ { "type": "interim", "title": "$$4\\cdot\\:2yy=8y^{2}$$", "input": "4\\cdot\\:2yy", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8yy" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$yy=\\:y^{1+1}$$" ], "result": "=8y^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bCD9BXAjMvvzFZ0XHk3and6GQqufR6tr2vPxOUv7H++oskUoGw3I/T4XJ4TyqVwsP8vQyhiD4JSfqjIvcQ7tisB+QJwCj80eajU0eXkOdNGPATzsdMJbIBTfKlGugMwN" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:y=4y$$", "input": "4\\cdot\\:1\\cdot\\:y", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4y" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7X+ZCeZ1HdwAXplX78369OejkVi15I8rBefLi4Iyt2wryZnY6JNhzvZPHzsebZpuF+mu8ChWaHA92hy+PE//GnL55skgg6xUxsA2+OqY57df" } }, { "type": "step", "result": "=-8y^{2}+4y" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P8rlQCZ4vUY+21PNyyoV9d13jtrSFDx+UNsawjlOjV321QDKUIiC37rwR8aif+LnyZnY6JNhzvZPHzsebZpuF4Jy8vaNLPDVWVFwZfiRrqqflQKmfe3SLw4Mglt+bdq9" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70XWLv2YsnGGRvhLiSccrraoksd6kLZ22p4dQOi2Y2q+J9HQfN3qOfAB4kBd9UOFDpA7clkQOa7zoZEIDggSG4WRLd2VwIqlBNByF6663syTD7/TDV+MdcR0GdN6+1FDtjNsJgrSbDO9QYEk3qmt0++IASZeFjDtawNGt9P21GJY=" } }, { "type": "step", "result": "4-8y^{2}+4y" } ], "meta": { "solvingClass": "Equations", "interimType": "Discriminant Title 1Eq" } }, { "type": "interim", "title": "$$4-8y^{2}+4y\\ge\\:0{\\quad:\\quad}-\\frac{1}{2}\\le\\:y\\le\\:1$$", "input": "4-8y^{2}+4y\\ge\\:0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "4-8y^{2}+4y\\ge\\:0", "result": "-2y^{2}+y+1\\ge\\:0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4}{4}-\\frac{8y^{2}}{4}+\\frac{4y}{4}\\ge\\:\\frac{0}{4}" }, { "type": "interim", "title": "Refine $$\\frac{4}{4}-\\frac{8y^{2}}{4}+\\frac{4y}{4}\\ge\\:\\frac{0}{4}:{\\quad}-2y^{2}+y+1\\ge\\:0$$", "input": "\\frac{4}{4}-\\frac{8y^{2}}{4}+\\frac{4y}{4}\\ge\\:\\frac{0}{4}", "result": "-2y^{2}+y+1\\ge\\:0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4}{4}-\\frac{8y^{2}}{4}+\\frac{4y}{4}:{\\quad}-2y^{2}+y+1$$", "input": "\\frac{4}{4}-\\frac{8y^{2}}{4}+\\frac{4y}{4}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "secondary": [ "$$\\frac{4}{4}=1$$" ], "result": "=1-\\frac{8y^{2}}{4}+\\frac{4y}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=1-2y^{2}+\\frac{4y}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{4}=1$$", "result": "=1-2y^{2}+y" }, { "type": "step", "primary": "Rewrite in standard form", "result": "=-2y^{2}+y+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\frac{0}{4}=0$$", "input": "\\frac{0}{4}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vptvTuZNa6XIeLtQYS+GG1XTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWB5NXi7bTNbXVXk8s7Avpa90jXmHMNyC4TYxqQbAxQjA=" } }, { "type": "step", "result": "-2y^{2}+y+1\\ge\\:0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Refine Specific 1Eq" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$-2y^{2}+y+1:{\\quad}-\\left(2y+1\\right)\\left(y-1\\right)$$", "input": "-2y^{2}+y+1", "result": "-\\left(2y+1\\right)\\left(y-1\\right)\\ge\\:0", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(2y^{2}-y-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$2y^{2}-y-1:{\\quad}\\left(2y+1\\right)\\left(y-1\\right)$$", "input": "2y^{2}-y-1", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2y^{2}-y-1", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-1,\\:c=-1$$", "$$u*v=-2,\\:u+v=-1$$" ] }, { "type": "interim", "title": "Factors of $$2:{\\quad}1,\\:2$$", "input": "2", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpp3lFwBgr08J1hDIhHaqLjwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJmyKVlUUECPEUZsSDkCFf8u/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$2$$", "result": "1,\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$2:{\\quad}-1,\\:-2$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-2,\\:$$check if $$u+v=-1$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-2:\\quad\\:u*v=-2,\\:u+v=-1\\quad\\Rightarrow\\quad\\:$$True", "secondary": [ "Check $$u=2,\\:v=-1:\\quad\\:u*v=-2,\\:u+v=1\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=-2" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2y^{2}+y\\right)+\\left(-2y-1\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwhtH/fSTucyIGBepDA0B7UUsjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKGpfcf6zmkh94vUJtR0x/7DXupxVJDtAMQSQ/ZMx+luaqSDBuXoUBQ5xeSNAoQTJGWnwiLOM+ZBhoGhL7orAkn3qWHLhID6nYJu+thEd/lleyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(2y^{2}+y\\right)+\\left(-2y-1\\right)" }, { "type": "interim", "title": "Factor out $$y\\:$$from $$2y^{2}+y:\\quad\\:y\\left(2y+1\\right)$$", "input": "2y^{2}+y", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{2}=yy$$" ], "result": "=2yy+y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(2y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-2y-1:\\quad\\:-\\left(2y+1\\right)$$", "input": "-2y-1", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(2y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7S6VlEk88VsbP8bQ6IPVhE5N1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwKwAmOXneCSPPkKyvDtUPmCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDuxuhvtrPBMHJcjkx5jJinPkdSMqJ1FDmbyi7zJ6JdQPpA=" } }, { "type": "step", "result": "=y\\left(2y+1\\right)-\\left(2y+1\\right)" }, { "type": "step", "primary": "Factor out common term $$2y+1$$", "result": "=\\left(2y+1\\right)\\left(y-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(2y+1\\right)\\left(y-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-\\left(2y+1\\right)\\left(y-1\\right)\\right)\\left(-1\\right)\\le\\:0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\left(2y+1\\right)\\left(y-1\\right)\\le\\:0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "-\\frac{1}{2}\\le\\:y\\le\\:1", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\left(2y+1\\right)\\left(y-1\\right)$$" }, { "type": "interim", "title": "Find the signs of $$2y+1$$", "steps": [ { "type": "interim", "title": "$$2y+1=0:{\\quad}y=-\\frac{1}{2}$$", "input": "2y+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2y+1=0", "result": "2y=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2y+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "2y=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "y=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$2y+1<0:{\\quad}y<-\\frac{1}{2}$$", "input": "2y+1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2y+1<0", "result": "2y<-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2y+1-1<0-1" }, { "type": "step", "primary": "Simplify", "result": "2y<-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "y<-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$2y+1>0:{\\quad}y>-\\frac{1}{2}$$", "input": "2y+1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2y+1>0", "result": "2y>-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2y+1-1>0-1" }, { "type": "step", "primary": "Simplify", "result": "2y>-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"result": "y>-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$y-1$$", "steps": [ { "type": "interim", "title": "$$y-1=0:{\\quad}y=1$$", "input": "y-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "y-1=0", "result": "y=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "y-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "y=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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[ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "y-1+1<0+1" }, { "type": "step", "primary": "Simplify", "result": "y<1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$y-1>0:{\\quad}y>1$$", "input": "y-1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "y-1>0", "result": "y>1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "y-1+1>0+1" }, { "type": "step", "primary": "Simplify", "result": "y>1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &y<-\\frac{1}{2}&y=-\\frac{1}{2}&-\\frac{1}{2}<y<1&y=1&y>1\\\\\\hline 2y+1&-&0&+&+&+\\\\\\hline y-1&-&-&-&0&+\\\\\\hline (2y+1)(y-1)&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$\\le\\:\\:0$$", "result": "y=-\\frac{1}{2}\\lor\\:-\\frac{1}{2}<y<1\\lor\\:y=1" }, { "type": "interim", "title": "Merge Overlapping Intervals", "input": "-\\frac{1}{2}\\le\\:y<1\\lor\\:y=1", "result": "-\\frac{1}{2}\\le\\:y\\le\\:1", "steps": [ { "type": "step", "primary": "The union of two intervals is the set of numbers which are in either interval<br/>$$y=-\\frac{1}{2}\\quad$$or$$\\quad\\:-\\frac{1}{2}<y<1$$", "image": "/images/interval?expression=%28y_%7B1%7D%3D-%5Cfrac%7B1%7D%7B2%7D%29%5Clor+%28-%5Cfrac%7B1%7D%7B2%7D%3Cy_%7B1%7D%3C1%29", "result": "-\\frac{1}{2}\\le\\:y<1" }, { "type": "step", "primary": "The union of two intervals is the set of numbers which are in either interval<br/>$$-\\frac{1}{2}\\le\\:y<1\\quad$$or$$\\quad\\:y=1$$", "image": "/images/interval?expression=%28-%5Cfrac%7B1%7D%7B2%7D%5Cle+y_%7B1%7D%3C1%29%5Clor+%28y_%7B1%7D%3D1%29", "result": "-\\frac{1}{2}\\le\\:y\\le\\:1" } ], "meta": { "interimType": "Merge Overlapping Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQIG5XqTlrmKAIafCjda27N2LBiFcA4ske/0vT4rq9yzhc8AJ98BYaQEm+J+qcjNzeHAfVxo1VcLEvsYw6coMwffgPuBLKtj6DifJlSlX4/THfbhNvmw+u6PuJsshco1Pm" } } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "step", "primary": "Check if the range interval endpoints are included" }, { "type": "interim", "title": "$$y=-\\frac{1}{2}{\\quad:\\quad}$$Included", "steps": [ { "type": "step", "primary": "Take the point $$y=-\\frac{1}{2}$$ and plug it into $$\\frac{2x+1}{x^{2}+2}=y$$", "result": "\\frac{2x+1}{x^{2}+2}=-\\frac{1}{2}" }, { "type": "interim", "title": "$$\\frac{2x+1}{x^{2}+2}=-\\frac{1}{2}{\\quad:\\quad}x=-2$$", "input": "\\frac{2x+1}{x^{2}+2}=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Cross multiply", "input": "\\frac{2x+1}{x^{2}+2}=-\\frac{1}{2}", "result": "\\left(2x+1\\right)\\cdot\\:2=-\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Apply fraction cross multiply: if $$\\frac{a}{b}=\\frac{c}{d}$$ then $$a\\cdot\\:d=b\\cdot\\:c$$", "result": "\\left(2x+1\\right)\\cdot\\:2=-\\left(x^{2}+2\\right)\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$-\\left(x^{2}+2\\right)\\cdot\\:1:{\\quad}-\\left(x^{2}+2\\right)$$", "input": "-\\left(x^{2}+2\\right)\\cdot\\:1", "result": "\\left(2x+1\\right)\\cdot\\:2=-\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$\\left(x^{2}+2\\right)\\cdot\\:1=\\left(x^{2}+2\\right)$$", "result": "=-\\left(x^{2}+2\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73WyAc3gnVaPrGOV+dkgrOGBFH3ZqAOJQyqKKX506iE3MwViaLUXkeD+JukROhWdjt2puP0GKG88tsuBXxXKXRYEFMST8lDZxn1Yq5HMKVTuiB03PGI/LJCjQKLZwNl+nUjZEJjEj6n+60/1WJdLemLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "interimType": "Equation Cross Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAjYcBSbJJKcO5nwp2W+NzBMAKPjwb0XFbSeuLXuKsbYVVPX2g4PtAv4VSQSFNN93S49bBNa1GGq8BdCQibmxQJymoP3ibkgfir2/qeDb/Gj6Im7nlbfLg3ZC7paZg44WGc0UAm97e3ZdFs70jM7AY+wWfs/98G5t1BK0SekTHW8CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Solve $$\\left(2x+1\\right)\\cdot\\:2=-\\left(x^{2}+2\\right):{\\quad}x=-2$$", "input": "\\left(2x+1\\right)\\cdot\\:2=-\\left(x^{2}+2\\right)", "steps": [ { "type": "interim", "title": "Expand $$\\left(2x+1\\right)\\cdot\\:2:{\\quad}4x+2$$", "input": "\\left(2x+1\\right)\\cdot\\:2", "steps": [ { "type": "step", "result": "=2\\left(2x+1\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=2x,\\:c=1$$" ], "result": "=2\\cdot\\:2x+2\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2x+2\\cdot\\:1:{\\quad}4x+2$$", "input": "2\\cdot\\:2x+2\\cdot\\:1", "result": "=4x+2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4x+2\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=4x+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ivc25tHYt2+GRUGonNHiY4nDdKucBi4LmQUhhh+vc1/oTCAC3hcNW1FUsqCI31nft2g6wAs+7JzewqfesYkp+O9sGZu5A1MXROmEpnxG69rsRwKOgJnDwVtTKH8A+1RzU1H2UXT9cyOxhYko5BX0/A==" } }, { "type": "interim", "title": "Expand $$-\\left(x^{2}+2\\right):{\\quad}-x^{2}-2$$", "input": "-\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(x^{2}\\right)-\\left(2\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-x^{2}-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VxtLaSJdXd/zwllqvDORDDN2s7Tu4QLkRRj00bh3VC51DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhmOLv/nGtLjwWC6N3796pNUl+uf6AzAndAjz3Vql+ewDQ==" } }, { "type": "step", "result": "4x+2=-x^{2}-2" }, { "type": "step", "primary": "Switch sides", "result": "-x^{2}-2=4x+2" }, { "type": "interim", "title": "Move $$2\\:$$to the left side", "input": "-x^{2}-2=4x+2", "result": "-x^{2}-4=4x", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "-x^{2}-2-2=4x+2-2" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}-4=4x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$4x\\:$$to the left side", "input": "-x^{2}-4=4x", "result": "-x^{2}-4-4x=0", "steps": [ { "type": "step", "primary": "Subtract $$4x$$ from both sides", "result": "-x^{2}-4-4x=4x-4x" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}-4-4x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-x^{2}-4x-4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-x^{2}-4x-4=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-4\\right)}}{2\\left(-1\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-1,\\:b=-4,\\:c=-4$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-4\\right)}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-4\\right)=0$$", "input": "\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-4\\right)", "result": "{x}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{0}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\left(-4\\right)^{2}-4\\cdot\\:1\\cdot\\:4" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-4\\right)^{2}=4^{2}$$" ], "result": "=4^{2}-4\\cdot\\:1\\cdot\\:4" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:4=16$$", "result": "=4^{2}-16" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16-16" }, { "type": "step", "primary": "Subtract the numbers: $$16-16=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ujB9qjgUyWfUzRTKfNZcRQY+8kJHsEPSNfPUw6mvOGvMwViaLUXkeD+JukROhWdjguVaeA0vkFtAR3gzqI9IZ5VWGZZCaGWfZZppDEVeDzm37urDJyZN0vivlFALhT0N" } }, { "type": "step", "result": "x=\\frac{-\\left(-4\\right)}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$\\frac{-\\left(-4\\right)}{2\\left(-1\\right)}=-2$$", "input": "\\frac{-\\left(-4\\right)}{2\\left(-1\\right)}", "result": "x=-2", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{4}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7w2s+0LBRUaYh/oyTz9oR9qIs/YioJczjhcNuzjHXFLqrju+5Z51e/ZZSD3gRHwjBy++q5TspDrdoyXuh0DUdTrtwsZ2WTt7bNTkti0EC8AoLw2D2HXl32XPYZVVlBKrc" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "x=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Solution exists, therefore $$y=-\\frac{1}{2}$$ is included in the range" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$y=1{\\quad:\\quad}$$Included", "steps": [ { "type": "step", "primary": "Take the point $$y=1$$ and plug it into $$\\frac{2x+1}{x^{2}+2}=y$$", "result": "\\frac{2x+1}{x^{2}+2}=1" }, { "type": "interim", "title": "$$\\frac{2x+1}{x^{2}+2}=1{\\quad:\\quad}x=1$$", "input": "\\frac{2x+1}{x^{2}+2}=1", "steps": [ { "type": "interim", "title": "Multiply both sides by $$x^{2}+2$$", "input": "\\frac{2x+1}{x^{2}+2}=1", "result": "2x+1=x^{2}+2", "steps": [ { "type": "step", "primary": "Multiply both sides by $$x^{2}+2$$", "result": "\\frac{2x+1}{x^{2}+2}\\left(x^{2}+2\\right)=1\\cdot\\:\\left(x^{2}+2\\right)" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x+1}{x^{2}+2}\\left(x^{2}+2\\right)=1\\cdot\\:\\left(x^{2}+2\\right)", "result": "2x+1=x^{2}+2", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x+1}{x^{2}+2}\\left(x^{2}+2\\right):{\\quad}2x+1$$", "input": "\\frac{2x+1}{x^{2}+2}\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(2x+1\\right)\\left(x^{2}+2\\right)}{x^{2}+2}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{2}+2$$", "result": "=2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7M44OTTnh0cgSx2vvkh4lDPLe3xbTjpN58v45dJoAn0YAlilG71elit3w1IBbYN0Pw2LDuVAu0yaIEG7Ywj2nYzq+Ng6EWuJZr7Vvl0wY8ilyhd7tjiG+GxQNxDvGkZUlgEmY3xmaQe71epl0I5lsuagktPvclYsNbQLF6Wo/cdKXL+c4UaAx9FzAfvS1SKj3" } }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\left(x^{2}+2\\right):{\\quad}x^{2}+2$$", "input": "1\\cdot\\:\\left(x^{2}+2\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x^{2}+2\\right)=\\left(x^{2}+2\\right)$$", "result": "=\\left(x^{2}+2\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x^{2}+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXK5qIWyOn7o62KuoBvXzuB8ejkVi15I8rBefLi4Iyt2wr7650LCtXjD19adwm/BgMFE3kCh3oevUunZ7/b0qFKBQwRgti3uQ2HhPD7HmoaimpBVHtasG0JuT9e6ygW07FPg==" } }, { "type": "step", "result": "2x+1=x^{2}+2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$2x+1=x^{2}+2:{\\quad}x=1$$", "input": "2x+1=x^{2}+2", "steps": [ { "type": "step", "primary": "Switch sides", "result": "x^{2}+2=2x+1" }, { "type": "interim", "title": "Move $$1\\:$$to the left side", "input": "x^{2}+2=2x+1", "result": "x^{2}+1=2x", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+2-1=2x+1-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}+1=2x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$2x\\:$$to the left side", "input": "x^{2}+1=2x", "result": "x^{2}+1-2x=0", "steps": [ { "type": "step", "primary": "Subtract $$2x$$ from both sides", "result": "x^{2}+1-2x=2x-2x" }, { "type": "step", "primary": "Simplify", "result": "x^{2}+1-2x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "x^{2}-2x+1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}-2x+1=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=-2,\\:c=1$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1=0$$", "input": "\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{0}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}-4\\cdot\\:1\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=2^{2}-4" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4-4" }, { "type": "step", "primary": "Subtract the numbers: $$4-4=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DPXAC421UjJhhGOo7DnYa25PLVy0IEOU1w5RhPMODOwgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09ngWpZrrqhM2PY7ADOWfoCdBY+wojvBUDQ0rdjc9Zpxevn7iF8w34il58z5Kk+cQTQ==" } }, { "type": "step", "result": "x=\\frac{-\\left(-2\\right)}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$\\frac{-\\left(-2\\right)}{2\\cdot\\:1}=1$$", "input": "\\frac{-\\left(-2\\right)}{2\\cdot\\:1}", "result": "x=1", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v228JUa8d3c6dzWBzNQkPLFZak7+g3aGbe9C49/LWHDNGoPE9TME3q+OPmgkv2RQHce6XHdcf2wFqSC/IQIp7Y+MAqL37zfLm0yQ/WPDUpmwkktQA3aNGhf00gW9Ij7KIrIyDjh72nkbsZF8nZ6f2A==" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Solution exists, therefore $$y=1$$ is included in the range" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Therefore the range is", "result": "-\\frac{1}{2}\\le\\:f\\left(x\\right)\\le\\:1" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Top 1Eq" } }, { "type": "interim", "title": "Axis interception points of $$\\frac{2x+1}{x^{2}+2}:\\quad\\:$$X Intercepts$$:\\:\\left(-\\frac{1}{2},\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:\\frac{1}{2}\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$\\frac{2x+1}{x^{2}+2}:{\\quad}\\left(-\\frac{1}{2},\\:0\\right)$$", "input": "\\frac{2x+1}{x^{2}+2}", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$\\frac{2x+1}{x^{2}+2}=0:{\\quad}x=-\\frac{1}{2}$$", "input": "\\frac{2x+1}{x^{2}+2}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "2x+1=0" }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2x+1=0", "result": "2x=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2x+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "2x=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Go+Y05pSq7WRPGDPAvfrgByOUtErWNRXbTqqjj69E3t1W+nD1X4ZhxKbfCIrI3vlRKvwpCsMm/dIDWKdnSUJtNIX39+dCbHwmNdtVu8mX2q8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAyDdf/fI5y88ZYxreEPCXm9MAKPjwb0XFbSeuLXuKsbYfpcF7HAGl5Qd9uLdeUchlWmQLNeZPPeALXLzkuhynB6va47ChjyjhYQNp84eixZmrt5g+KB4e7b6i2FR+k9P7Pb6UxNL6vxesNluH3rG2Je63PoO3Pr0pBdM+SW4qgtDmdy0/VI8pzLrUrY9XFIAYHyA9VUqucCMlGr81hxrGphrOuqqP5zx821CyjzgB1MgZ52Ot4gtyhgxRXuC+2sLq+6aphXEe7KE/W0X5BIzZ7reRzM3vynqxJKIg3jtF9ZCY+dRwkanZaJUwyYScC1TCzvlUs9ssN4GudQG+VOP3Gnyt8LFQm5UpsbsZ5ifTkhFGRpIJqMaDZWJwDpG5SWwcsd7ZG6RIQIR6lm1Ekl71S7RSpN33oxZMojoqvYhvSJADoM5fP+aNEEBI7cZqWPuM+cqeFNvkW4u+eykK6QpWbUbqgCeJqTEq1trtv68VFB6w==" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=-1", "result": "x=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{-1}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(-\\frac{1}{2},\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7Pb6UxNL6vxesNluH3rG2JePrq1bFImqxvf7VMH9ehqlGkgwiJs5bIu0pdHx4WbtBlPa6cQLTEyKlDipFDOAKIpFkDvK1UMnwuqxsrVOXirujACj48G9FxW0nri17irG2GPATzsdMJbIBTfKlGugMwN" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$\\frac{2x+1}{x^{2}+2}:{\\quad}\\left(0,\\:\\frac{1}{2}\\right)$$", "input": "\\frac{2x+1}{x^{2}+2}", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=\\frac{2\\cdot\\:0+1}{0^{2}+2}:{\\quad}y=\\frac{1}{2}$$", "input": "y=\\frac{2\\cdot\\:0+1}{0^{2}+2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\cdot\\:0+1}{0^{2}+2}:{\\quad}\\frac{1}{2}$$", "input": "\\frac{2\\cdot\\:0+1}{0^{2}+2}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=\\frac{2\\cdot\\:0+1}{0+2}" }, { "type": "interim", "title": "$$2\\cdot\\:0+1=1$$", "input": "2\\cdot\\:0+1", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0+1" }, { "type": "step", "primary": "Add the numbers: $$0+1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lLk8RjdIVjV71gT+Xwch196GQqufR6tr2vPxOUv7H++P6Ubiv/bIrpol3G9QIK7h1SvY/eJGzvEmlW7hoPETDkiEjEdkNi+fL5poUC3HY70=" } }, { "type": "step", "result": "=\\frac{1}{0+2}" }, { "type": "step", "primary": "Add the numbers: $$0+2=2$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wwMXmpSw8uHSAcJv7PDwnIN5mdW5WcL8mdV714+nlJV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAln+sIiWZ2/rqewsnJFNPZ9BN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+Upuofd1oss9YoZoKvZkJJQ/Ui6r8JFyFa8EmhS0Pj1C" } }, { "type": "step", "result": "y=\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:\\frac{1}{2}\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfkYIe2KSA99imKxFofRuiuRTDzCWaLrWv5cU4/UsvpgYH9kVsi5PE1PJmCjCy4rHJ2O5FEYlptVV2+TVuQDnXvIrB0XG+mDEsGJmz2jn1ZFzSm5Ki9XyQY5tWThYwJ1pd+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(-\\frac{1}{2},\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:\\frac{1}{2}\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Asymptotes of $$\\frac{2x+1}{x^{2}+2}:\\quad\\:$$Horizontal$$:\\:y=0$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{2x+1}{x^{2}+2}:{\\quad}$$None", "input": "\\frac{2x+1}{x^{2}+2}", "steps": [ { "type": "definition", "title": "Vertical asymptotes of rational Functions", "text": "For rational functions, the vertical asymptotes are the undefined points, also known as the zeros of the denominator, of the simplified function." }, { "type": "step", "primary": "The function $$\\frac{2x+1}{x^{2}+2}:\\:$$True for all $$x\\in\\mathbb{R}\\:$$has no undefined points" }, { "type": "step", "result": "\\mathrm{No\\:vertical\\:asymptotes}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeJtEhphkHIS+T3ezFbRylFso3oe/oyhMy2+1TQhDBd2f2eADB+dAKOx4BAaVcUG+vkD1WOHaLAbMOr6c56Q4EkUnnY63iC3KGDFFe4L7awurxK458vXQ46K82kIiIy6Grg=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{2x+1}{x^{2}+2}:{\\quad}y=0$$", "input": "\\frac{2x+1}{x^{2}+2}", "steps": [ { "type": "definition", "title": "Horizontal asymptotes of rational functions", "text": "If denominator's degree > numerator's degree, the x-axis is the horizontal asymptote.<br/>If the degrees are equal, there is an horizontal asymptote: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$<br/>Otherwise, there is no horizontal asymptote." }, { "type": "step", "primary": "The degree of the numerator$$=1.\\:$$The degree of the denominator$$=2$$", "secondary": [ "Denominator's degree > numerator's degree. Therefore, the horizontal asymptote is the x-axis" ] }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MtCQmRPbaSWV5kXIOrW1iHIWFYzcA4oaZR79v7JdxE/FjEnStUIOCBezKqcGh3V47Bl54axjF89+pySUHsh45mRT2gmDw3ilFxbV10bWZ5MBeHDeCsqyidvL+aXp5vCyq8=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{2x+1}{x^{2}+2}:{\\quad}$$None", "input": "\\frac{2x+1}{x^{2}+2}", "steps": [ { "type": "definition", "title": "Slant asymptotes of rational functions", "text": "If numerator's degree = 1 + denominator's degree, there is a slant asymptote of the form: y=mx+b.<br/>Otherwise there is no slant asymptote" }, { "type": "step", "primary": "The degree of the numerator$$=1.\\:$$The degree of the denominator$$=2$$", "secondary": [ "Numerator's degree $$\\neq\\:$$ 1 + denominator's degree" ] }, { "type": "step", "primary": "Therefore there is no slant asymptote" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KGMi33OHHY/rSfQuuka/CjHfm2MhbpuXfQjeVbgj1bl/gQUxJPyUNnGfVirkcwpVO9texQWP30zBG2LsG2smZ5LBuGK/MPjtR4jogh8jhqHIvVkWgsc85J7t6Iim+Vb6myS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "\\mathrm{Horizontal}:\\:y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Function Asymptotes Top 2Eq" } }, { "type": "interim", "title": "Extreme Points of $$\\frac{2x+1}{x^{2}+2}:{\\quad}$$Minimum$$\\left(-2,\\:-\\frac{1}{2}\\right),\\:$$Maximum$$\\left(1,\\:1\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{2x+1}{x^{2}+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+1\\right)\\left(x^{2}+2\\right)-\\frac{d}{dx}\\left(x^{2}+2\\right)\\left(2x+1\\right)}{\\left(x^{2}+2\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x+1\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(2x\\right)+\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "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": "=2+0" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+2\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{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(x^{2}\\right)+\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "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": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{2\\left(x^{2}+2\\right)-2x\\left(2x+1\\right)}{\\left(x^{2}+2\\right)^{2}}" }, { "type": "interim", "title": "Expand $$2\\left(x^{2}+2\\right)-2x\\left(2x+1\\right):{\\quad}-2x^{2}-2x+4$$", "input": "2\\left(x^{2}+2\\right)-2x\\left(2x+1\\right)", "result": "=\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}", "steps": [ { "type": "interim", "title": "Expand $$2\\left(x^{2}+2\\right):{\\quad}2x^{2}+4$$", "input": "2\\left(x^{2}+2\\right)", "result": "=2x^{2}+4-2x\\left(2x+1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=x^{2},\\:c=2$$" ], "result": "=2x^{2}+2\\cdot\\:2", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2x^{2}+4" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s781TTQ5DKjvPKHmpiVekdvyAn9lkDfZkicUGkO3EF+IoVEXtDWYqdfh5vws5SjuzLIvsJv/VpmBmOllN+wMnDXfxtgDRvpXZFJM5ns6z004P2/YRbooLH6JXZok6RxWRQ" } }, { "type": "interim", "title": "Expand $$-2x\\left(2x+1\\right):{\\quad}-4x^{2}-2x$$", "input": "-2x\\left(2x+1\\right)", "result": "=2x^{2}+4-4x^{2}-2x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-2x,\\:b=2x,\\:c=1$$" ], "result": "=-2x\\cdot\\:2x+\\left(-2x\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2\\cdot\\:2xx-2\\cdot\\:1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:2xx-2\\cdot\\:1\\cdot\\:x:{\\quad}-4x^{2}-2x$$", "input": "-2\\cdot\\:2xx-2\\cdot\\:1\\cdot\\:x", "result": "=-4x^{2}-2x", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:2xx=4x^{2}$$", "input": "2\\cdot\\:2xx", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4xx" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=4x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OCdoueT0//6uVMt1K/MfC96GQqufR6tr2vPxOUv7H+8SfhH/L0ue+vNHkCC+SCamP8vQyhiD4JSfqjIvcQ7tisZHQ8+zxICZdm4BqTGbadaPATzsdMJbIBTfKlGugMwN" } }, { "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": "=-4x^{2}-2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xe+Z7o6Iraysrv6F4Vpz2N13jtrSFDx+UNsawjlOjV14+Au7R4CYwXPPv/1fdSL72GIoxg2Jpr0LaZPQ02JWI4Jy8vaNLPDVWVFwZfiRrqr4OMf9z8lpl5V6yH4/2u+c" } }, { "type": "interim", "title": "Simplify $$2x^{2}+4-4x^{2}-2x:{\\quad}-2x^{2}-2x+4$$", "input": "2x^{2}+4-4x^{2}-2x", "result": "=-2x^{2}-2x+4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2x^{2}-4x^{2}-2x+4" }, { "type": "step", "primary": "Add similar elements: $$2x^{2}-4x^{2}=-2x^{2}$$", "result": "=-2x^{2}-2x+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kVNc/k5GnMWRJ7/RzaWuE9M+g6GD+Q1DUqZu5yyuRu5wkKGJWEPFPk38sdJMsyPI3VNokMBXn7TWRqV9KnXSJqN6Hv6MoTMtvtU0IQwXdn/3klWdMfBvel+KEKXV1FpY2lFCPVfIFb0/ExIX8BFIXmHE33NZtV6QHzGTclQ6fVo=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Decreasing$$:-\\infty\\:<x<-2,\\:$$Increasing$$:-2<x<1,\\:$$Decreasing$$:1<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}x=-2,\\:x=1$$", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its derivative is zero or undefined" }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=0:{\\quad}x=-2,\\:x=1$$", "input": "\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "-2x^{2}-2x+4=0" }, { "type": "interim", "title": "Solve $$-2x^{2}-2x+4=0:{\\quad}x=-2,\\:x=1$$", "input": "-2x^{2}-2x+4=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-2x^{2}-2x+4=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}}{2\\left(-2\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-2,\\:b=-2,\\:c=4$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}}{2\\left(-2\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}=6$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:2\\cdot\\:4}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}+4\\cdot\\:2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:4=32$$", "result": "=\\sqrt{2^{2}+32}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+32}" }, { "type": "step", "primary": "Add the numbers: $$4+32=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTkyNvvFb3YApY88JKBDgydwAlilG71elit3w1IBbYN0POfzEJvy01qFtiDDf1FBy4KN6Hv6MoTMtvtU0IQwXdn9gemVBIzJhCSDpYD1Ky1x2Rtr5bbogjnqwl8ACFn8KGSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)},\\:{x}_{2}=\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)}:{\\quad}-2$$", "input": "\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{2+6}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\frac{8}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{8}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{8}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFqNWcG7GsnnezBHiIMWJNdzTQFcXvE5M6pxvFYiUr51g99dC9fj9sg0EHzBIRDRwJRuKB0+skz49yokD/nm0vAKCfFHLER2iRFCuUUu6VNdjZf53lsfXPAzKZQTtCk9vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}:{\\quad}1$$", "input": "\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{2-6}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$2-6=-4$$", "result": "=\\frac{-4}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-4}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{4}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72LopWbBMxyRtD0Gow7zUstdzTQFcXvE5M6pxvFYiUr51g99dC9fj9sg0EHzBIRDR+8ZDu8iF4MSewt4yms1lIb3Tcan7wFkSuOUEIM4ZHqH0sdj/x4QjVXxb0xKhtjFEJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=-2,\\:x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-2,\\:x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=-2,\\:x=1" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}-2<x<1$$", "input": "\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}>0", "steps": [ { "type": "interim", "title": "Factor $$\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}:{\\quad}\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}$$", "input": "\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}", "result": "\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}>0", "steps": [ { "type": "interim", "title": "Factor $$-2x^{2}-2x+4:{\\quad}-2\\left(x-1\\right)\\left(x+2\\right)$$", "input": "-2x^{2}-2x+4", "result": "=\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}", "steps": [ { "type": "interim", "title": "Factor out common term $$-2:{\\quad}-2\\left(x^{2}+x-2\\right)$$", "input": "-2x^{2}-2x+4", "steps": [ { "type": "step", "primary": "Rewrite $$4$$ as $$2\\cdot\\:2$$", "result": "=-2x^{2}-2x+2\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-2$$", "result": "=-2\\left(x^{2}+x-2\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=-2\\left(x^{2}+x-2\\right)" }, { "type": "interim", "title": "Factor $$x^{2}+x-2:{\\quad}\\left(x-1\\right)\\left(x+2\\right)$$", "input": "x^{2}+x-2", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+x-2", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=1,\\:c=-2$$", "$$u*v=-2,\\:u+v=1$$" ] }, { "type": "interim", "title": "Factors of $$2:{\\quad}1,\\:2$$", "input": "2", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpp3lFwBgr08J1hDIhHaqLjwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJmyKVlUUECPEUZsSDkCFf8u/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$2$$", "result": "1,\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$2:{\\quad}-1,\\:-2$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-2,\\:$$check if $$u+v=1$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-2:\\quad\\:u*v=-2,\\:u+v=-1\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=-1:\\quad\\:u*v=-2,\\:u+v=1\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=2,\\:v=-1" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(2x-2\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwn30q2MIJp/1sy1h0ZoW6o4HjZ0JmeAC3ZSEmMxWRNYu9GZPFaw2ewNA2aBnABa7Ku/snSOMzFBF/81Glop5NPlBgI/uz/WXQFrWuIqLHSyLITz6WHZx9UH0EGxKBGtUEBBOfyC7Me/gy/zVrVb5jy6lM1XdHV6wouRb0ZBuz4bhG4HL8yxsY3faiRjEFSmIgQ==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(2x-2\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$2\\:$$from $$2x-2:\\quad\\:2\\left(x-1\\right)$$", "input": "2x-2", "steps": [ { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7moABaoXIXq3E1MypH32OrsevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G18aJzGOjaRVSiBHKVuIO6fKN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17Xs0+6jPTos8h/1RIBPummW+tl6WFd7TeP+ieu4LzFGw2I=" } }, { "type": "step", "result": "=x\\left(x-1\\right)+2\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x+2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-2\\left(x-1\\right)\\left(x+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-2\\left(x-1\\right)\\left(x+2\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+2\\right)^{2}}<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}<0", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{\\frac{2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}}{2}<\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}<0" }, { "type": "interim", "title": "Identify the intervals", "result": "-2<x<1", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}$$" }, { "type": "interim", "title": "Find the signs of $$x-1$$", "steps": [ { "type": "interim", "title": "$$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-1<0:{\\quad}x<1$$", "input": "x-1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1<0", "result": "x<1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1<0+1" }, { "type": "step", "primary": "Simplify", "result": "x<1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-1>0:{\\quad}x>1$$", "input": "x-1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1>0", "result": "x>1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1>0+1" }, { "type": "step", "primary": "Simplify", "result": "x>1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cjSCGWDmRYpKcWh26iDbVJN1pXT08zEQpn0WJ6CFMXD+Bj/OTEQM9+GhM5gnqbimIv3pLYfKyQbKCshhkaZHUc+jHstXtCY+2iLEdcYc6HuJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SE18XPPEg1YN4TPF5UodL2dnbHrNlGWc0nrdBwYSEYpbgQexMdLOwGjjdSV48SZP4v63Er/yV7HiwKr+Sdi2DC0puSovV8kGObVk4WMCdaXfLOhY7AFpgLohmjnNvGn518esWE1fF3rWh+qTXbIWpC+mQLNeZPPeALXLzkuhynB6Htx/gc2U4DTHXTUgNfzX2ahnQUvKI5VdMCgmO+TQs8XrQA/fY/1iZ7Jvhy4SNw2bE8zk56OYcMc5hnjwQpDx+UJCZE9tpJZXmRcg6tbWIcjwGmcxBL0FhjlgXVX33fhFFZPLdk8I4fs46fXZWE3KPxoMEmyHHeSYz/3dSrgm1veMw0n9r4f4vt7OHDv6CsGctaysPu9leNS4MXgjnnClznyL7URSLQlHzu9g25SQ76+moN+fZySAkjcUdbHZIiWA8LfSxJ+0AgVLpCSnLX0iSg/92zdqVBwRPhoObfbv3ebps4ZkzyezUnQsdjwVywvCRs5BUIMPXPRLS1S6F6PH3A==" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x+2$$", "steps": [ { "type": "interim", "title": "$$x+2=0:{\\quad}x=-2$$", "input": "x+2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x+2=0", "result": "x=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2<0-2" }, { "type": "step", "primary": "Simplify", "result": "x<-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x+2>0:{\\quad}x>-2$$", "input": "x+2>0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x+2>0", "result": "x>-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2>0-2" }, { "type": "step", "primary": "Simplify", "result": "x>-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7abxPFJKP3h67pmT8Dtekq5N1pXT08zEQpn0WJ6CFMXCzpow5G3NsgikccJImBCG8Iv3pLYfKyQbKCshhkaZHUUEaUpDLp6RmPP2lch8bJMqyiHU3VsxZ0Jvlgz1Dh/2W0zIpeqOKDWh7ydUbxgKhdXr3g+Hzr2XIOhBR2yznfmuo4e7l7yHtELyi7p5lEpRAKbkqL1fJBjm1ZOFjAnWl3yuc6uQXhPp7EgKL2RPRFT9CQmRPbaSWV5kXIOrW1iHIrY8ccVVfQ3KXYiO0ixEVSZzWWBIJEpoyG2857ckr7tiedjreILcoYMUV7gvtrC6vw6gEmJQ3lqr9coTvcg7iX+RioUfoGcTI5/RhtFL6RIKgoCvSjIXn9FpPM0l9uPIZ0f+jPL7LepBIKDtSUDhE4jACj48G9FxW0nri17irG2Hb7Pg8BWb7ciE+5DPCFBEsXJSGXy80Sd20ZLFLNZTKePIV5/jyXxXN1IdSTj6+cI9Imd2wdAYFLcEza9yp2aRVsNeBZgFdOkEsIHe9+v51bOZ08YYV/yTKcpM2SOid1ayhqMTMOtusdz6jZGZSsSW0gQUxJPyUNnGfVirkcwpVO8j2e4tci5bu9rJwUvH0IBGxHTToAjTqRFQ8/ptr/mJBn6h/ScAAB1sKAGmZ7yAM5g==" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$\\left(x^{2}+2\\right)^{2}$$", "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(x^{2}+2\\right)^{2}:{\\quad}$$No Solution", "input": "\\left(x^{2}+2\\right)^{2}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+2=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x^{2}+2=0", "result": "x^{2}=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x^{2}+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &x<-2&x=-2&-2<x<1&x=1&x>1\\\\\\hline x-1&-&-&-&0&+\\\\\\hline x+2&-&0&+&+&+\\\\\\hline (x^{2}+2)^{2}&+&+&+&+&+\\\\\\hline \\frac{(x-1)(x+2)}{(x^{2}+2)^{2}}&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$<\\:0$$", "result": "-2<x<1" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}x<-2\\lor\\:x>1$$", "input": "\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}<0", "steps": [ { "type": "interim", "title": "Factor $$\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}:{\\quad}\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}$$", "input": "\\frac{-2x^{2}-2x+4}{\\left(x^{2}+2\\right)^{2}}", "result": "\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}<0", "steps": [ { "type": "interim", "title": "Factor $$-2x^{2}-2x+4:{\\quad}-2\\left(x-1\\right)\\left(x+2\\right)$$", "input": "-2x^{2}-2x+4", "result": "=\\frac{-2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}", "steps": [ { "type": "interim", "title": "Factor out common term $$-2:{\\quad}-2\\left(x^{2}+x-2\\right)$$", "input": "-2x^{2}-2x+4", "steps": [ { "type": "step", "primary": "Rewrite $$4$$ as $$2\\cdot\\:2$$", "result": "=-2x^{2}-2x+2\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-2$$", "result": "=-2\\left(x^{2}+x-2\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=-2\\left(x^{2}+x-2\\right)" }, { "type": "interim", "title": "Factor $$x^{2}+x-2:{\\quad}\\left(x-1\\right)\\left(x+2\\right)$$", "input": "x^{2}+x-2", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+x-2", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=1,\\:c=-2$$", "$$u*v=-2,\\:u+v=1$$" ] }, { "type": "interim", "title": "Factors of $$2:{\\quad}1,\\:2$$", "input": "2", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpp3lFwBgr08J1hDIhHaqLjwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJmyKVlUUECPEUZsSDkCFf8u/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$2$$", "result": "1,\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$2:{\\quad}-1,\\:-2$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-2,\\:$$check if $$u+v=1$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-2:\\quad\\:u*v=-2,\\:u+v=-1\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=-1:\\quad\\:u*v=-2,\\:u+v=1\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=2,\\:v=-1" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(2x-2\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwn30q2MIJp/1sy1h0ZoW6o4HjZ0JmeAC3ZSEmMxWRNYu9GZPFaw2ewNA2aBnABa7Ku/snSOMzFBF/81Glop5NPlBgI/uz/WXQFrWuIqLHSyLITz6WHZx9UH0EGxKBGtUEBBOfyC7Me/gy/zVrVb5jy6lM1XdHV6wouRb0ZBuz4bhG4HL8yxsY3faiRjEFSmIgQ==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(2x-2\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$2\\:$$from $$2x-2:\\quad\\:2\\left(x-1\\right)$$", "input": "2x-2", "steps": [ { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7moABaoXIXq3E1MypH32OrsevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G18aJzGOjaRVSiBHKVuIO6fKN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17Xs0+6jPTos8h/1RIBPummW+tl6WFd7TeP+ieu4LzFGw2I=" } }, { "type": "step", "result": "=x\\left(x-1\\right)+2\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x+2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-2\\left(x-1\\right)\\left(x+2\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-2\\left(x-1\\right)\\left(x+2\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+2\\right)^{2}}>0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}>0", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{\\frac{2\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}}{2}>\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}>0" }, { "type": "interim", "title": "Identify the intervals", "result": "x<-2\\lor\\:x>1", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{\\left(x-1\\right)\\left(x+2\\right)}{\\left(x^{2}+2\\right)^{2}}$$" }, { "type": "interim", "title": "Find the signs of $$x-1$$", "steps": [ { "type": "interim", "title": "$$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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