asymptotes of (3x^2-18x+24)/(x^2-4x)

{ "query": { "display": "asymptotes $$\\frac{3x^{2}-18x+24}{x^{2}-4x}$$", "symbolab_question": "FUNCTION#asymptotes \\frac{3x^{2}-18x+24}{x^{2}-4x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=0,\\mathrm{Horizontal}: y=3", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:\\quad\\:$$Vertical$$:\\:x=0,\\:$$Horizontal$$:\\:y=3$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}x=0$$", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}\\frac{3\\left(x-2\\right)}{x}$$", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor $$3x^{2}-18x+24:{\\quad}3\\left(x-2\\right)\\left(x-4\\right)$$", "input": "3x^{2}-18x+24", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor out common term $$3:{\\quad}3\\left(x^{2}-6x+8\\right)$$", "input": "3x^{2}-18x+24", "steps": [ { "type": "step", "primary": "Rewrite $$24$$ as $$3\\cdot\\:8$$", "secondary": [ "Rewrite $$18$$ as $$3\\cdot\\:6$$" ], "result": "=3x^{2}-3\\cdot\\:6x+3\\cdot\\:8" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x^{2}-6x+8\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=3\\left(x^{2}-6x+8\\right)" }, { "type": "interim", "title": "Factor $$x^{2}-6x+8:{\\quad}\\left(x-2\\right)\\left(x-4\\right)$$", "input": "x^{2}-6x+8", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-6x+8", "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": [ "$$a=1,\\:b=-6,\\:c=8$$", "$$u*v=8,\\:u+v=-6$$" ] }, { "type": "interim", "title": "Factors of $$8:{\\quad}1,\\:2,\\:4,\\:8$$", "input": "8", "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 $$8:{\\quad}2,\\:2,\\:2$$", "input": "8", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlrHmICBBU/V4v9mDHayw0Dwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJiL6o6RfUkqI5hqGsUSBhPK/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Multiply the prime factors of $$8:{\\quad}4$$", "result": "4", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$" }, { "type": "step", "result": "4" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$8\\:$$ itself", "result": "1,\\:8" }, { "type": "step", "primary": "The factors of $$8$$", "result": "1,\\:2,\\:4,\\:8" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$8:{\\quad}-1,\\:-2,\\:-4,\\:-8$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-8" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=8,\\:$$check if $$u+v=-6$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=8:\\quad\\:u*v=8,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=4:\\quad\\:u*v=8,\\:u+v=6\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-8:\\quad\\:u*v=8,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-2,\\:v=-4:\\quad\\:u*v=8,\\:u+v=-6\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-2,\\:v=-4" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwvWUYezO74qahsqRLThu9ZosjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKE7E2ceIHXfILXWwVIHhsSpp2dZ5nv2+jXnyAH/SElMZKSDBuXoUBQ5xeSNAoQTJGWnwiLOM+ZBhoGhL7orAkn3qWHLhID6nYJu+thEd/lleyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-2x:\\quad\\:x\\left(x-2\\right)$$", "input": "x^{2}-2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-4\\:$$from $$-4x+8:\\quad\\:-4\\left(x-2\\right)$$", "input": "-4x+8", "steps": [ { "type": "step", "primary": "Rewrite $$8$$ as $$4\\cdot\\:2$$", "result": "=-4x+4\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-4$$", "result": "=-4\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(x-2\\right)-4\\left(x-2\\right)" }, { "type": "step", "primary": "Factor out common term $$x-2$$", "result": "=\\left(x-2\\right)\\left(x-4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=3\\left(x-2\\right)\\left(x-4\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-4\\right)$$", "input": "x^{2}-4x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-4\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x\\left(x-4\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$x-4$$", "result": "=\\frac{3\\left(x-2\\right)}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "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": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{3\\left(x-2\\right)}{x}$$ and compare to zero" }, { "type": "step", "result": "x=0" }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKbf0MF7KrFKEL+eIrIWcJ0zX2k1vfbbThOURnVjhhi9nql8XXPq6bNQlMm+36iNhnt8tzMDLA2SXm8vTlWgnSCwbhivzD47UeI6IIfI4ahyGO1ol7VP865HzATK+4JqG5u4VgqncRtbfbn4CeGsyZW" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}y=3$$", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}\\frac{3\\left(x-2\\right)}{x}$$", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor $$3x^{2}-18x+24:{\\quad}3\\left(x-2\\right)\\left(x-4\\right)$$", "input": "3x^{2}-18x+24", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor out common term $$3:{\\quad}3\\left(x^{2}-6x+8\\right)$$", "input": "3x^{2}-18x+24", "steps": [ { "type": "step", "primary": "Rewrite $$24$$ as $$3\\cdot\\:8$$", "secondary": [ "Rewrite $$18$$ as $$3\\cdot\\:6$$" ], "result": "=3x^{2}-3\\cdot\\:6x+3\\cdot\\:8" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x^{2}-6x+8\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=3\\left(x^{2}-6x+8\\right)" }, { "type": "interim", "title": "Factor $$x^{2}-6x+8:{\\quad}\\left(x-2\\right)\\left(x-4\\right)$$", "input": "x^{2}-6x+8", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-6x+8", "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": [ "$$a=1,\\:b=-6,\\:c=8$$", "$$u*v=8,\\:u+v=-6$$" ] }, { "type": "interim", "title": "Factors of $$8:{\\quad}1,\\:2,\\:4,\\:8$$", "input": "8", "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 $$8:{\\quad}2,\\:2,\\:2$$", "input": "8", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlrHmICBBU/V4v9mDHayw0Dwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJiL6o6RfUkqI5hqGsUSBhPK/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Multiply the prime factors of $$8:{\\quad}4$$", "result": "4", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$" }, { "type": "step", "result": "4" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$8\\:$$ itself", "result": "1,\\:8" }, { "type": "step", "primary": "The factors of $$8$$", "result": "1,\\:2,\\:4,\\:8" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$8:{\\quad}-1,\\:-2,\\:-4,\\:-8$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-8" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=8,\\:$$check if $$u+v=-6$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=8:\\quad\\:u*v=8,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=4:\\quad\\:u*v=8,\\:u+v=6\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-8:\\quad\\:u*v=8,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-2,\\:v=-4:\\quad\\:u*v=8,\\:u+v=-6\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-2,\\:v=-4" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwvWUYezO74qahsqRLThu9ZosjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKE7E2ceIHXfILXWwVIHhsSpp2dZ5nv2+jXnyAH/SElMZKSDBuXoUBQ5xeSNAoQTJGWnwiLOM+ZBhoGhL7orAkn3qWHLhID6nYJu+thEd/lleyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-2x:\\quad\\:x\\left(x-2\\right)$$", "input": "x^{2}-2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-4\\:$$from $$-4x+8:\\quad\\:-4\\left(x-2\\right)$$", "input": "-4x+8", "steps": [ { "type": "step", "primary": "Rewrite $$8$$ as $$4\\cdot\\:2$$", "result": "=-4x+4\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-4$$", "result": "=-4\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(x-2\\right)-4\\left(x-2\\right)" }, { "type": "step", "primary": "Factor out common term $$x-2$$", "result": "=\\left(x-2\\right)\\left(x-4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=3\\left(x-2\\right)\\left(x-4\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-4\\right)$$", "input": "x^{2}-4x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-4\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x\\left(x-4\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$x-4$$", "result": "=\\frac{3\\left(x-2\\right)}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "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$$=1$$", "secondary": [ "The degrees are equal, the horizontal asymptote is: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$" ] }, { "type": "step", "primary": "Numerator's leading coefficient$$=3,\\:$$Denominator's leading coefficient$$=1$$", "result": "y=\\frac{3}{1}" }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=3" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MvJnTgm4jqAdr/1nVGldQ0sWWeJmDhhV2sruL7330Ob4/C30sSftAIFS6Qkpy19IkpUSrXuoR8wrTnQA8tZS/rDrBLih6IkUKnIZ24PAGQcmMwyWnZPijKEhcu0w296HvQKaOde6WT0kherXAWMTtxZ" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}$$None", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x^{2}-18x+24}{x^{2}-4x}:{\\quad}\\frac{3\\left(x-2\\right)}{x}$$", "input": "\\frac{3x^{2}-18x+24}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor $$3x^{2}-18x+24:{\\quad}3\\left(x-2\\right)\\left(x-4\\right)$$", "input": "3x^{2}-18x+24", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x^{2}-4x}", "steps": [ { "type": "interim", "title": "Factor out common term $$3:{\\quad}3\\left(x^{2}-6x+8\\right)$$", "input": "3x^{2}-18x+24", "steps": [ { "type": "step", "primary": "Rewrite $$24$$ as $$3\\cdot\\:8$$", "secondary": [ "Rewrite $$18$$ as $$3\\cdot\\:6$$" ], "result": "=3x^{2}-3\\cdot\\:6x+3\\cdot\\:8" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x^{2}-6x+8\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=3\\left(x^{2}-6x+8\\right)" }, { "type": "interim", "title": "Factor $$x^{2}-6x+8:{\\quad}\\left(x-2\\right)\\left(x-4\\right)$$", "input": "x^{2}-6x+8", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-6x+8", "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": [ "$$a=1,\\:b=-6,\\:c=8$$", "$$u*v=8,\\:u+v=-6$$" ] }, { "type": "interim", "title": "Factors of $$8:{\\quad}1,\\:2,\\:4,\\:8$$", "input": "8", "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 $$8:{\\quad}2,\\:2,\\:2$$", "input": "8", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlrHmICBBU/V4v9mDHayw0Dwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJiL6o6RfUkqI5hqGsUSBhPK/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Multiply the prime factors of $$8:{\\quad}4$$", "result": "4", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$" }, { "type": "step", "result": "4" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$8\\:$$ itself", "result": "1,\\:8" }, { "type": "step", "primary": "The factors of $$8$$", "result": "1,\\:2,\\:4,\\:8" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$8:{\\quad}-1,\\:-2,\\:-4,\\:-8$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-8" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=8,\\:$$check if $$u+v=-6$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=8:\\quad\\:u*v=8,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=4:\\quad\\:u*v=8,\\:u+v=6\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-8:\\quad\\:u*v=8,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-2,\\:v=-4:\\quad\\:u*v=8,\\:u+v=-6\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-2,\\:v=-4" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwvWUYezO74qahsqRLThu9ZosjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKE7E2ceIHXfILXWwVIHhsSpp2dZ5nv2+jXnyAH/SElMZKSDBuXoUBQ5xeSNAoQTJGWnwiLOM+ZBhoGhL7orAkn3qWHLhID6nYJu+thEd/lleyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\left(x^{2}-2x\\right)+\\left(-4x+8\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-2x:\\quad\\:x\\left(x-2\\right)$$", "input": "x^{2}-2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-4\\:$$from $$-4x+8:\\quad\\:-4\\left(x-2\\right)$$", "input": "-4x+8", "steps": [ { "type": "step", "primary": "Rewrite $$8$$ as $$4\\cdot\\:2$$", "result": "=-4x+4\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-4$$", "result": "=-4\\left(x-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(x-2\\right)-4\\left(x-2\\right)" }, { "type": "step", "primary": "Factor out common term $$x-2$$", "result": "=\\left(x-2\\right)\\left(x-4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=3\\left(x-2\\right)\\left(x-4\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-4\\right)$$", "input": "x^{2}-4x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-4\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{3\\left(x-2\\right)\\left(x-4\\right)}{x\\left(x-4\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$x-4$$", "result": "=\\frac{3\\left(x-2\\right)}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "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$$=1$$", "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/tmv96KKA8HMGdiIVS99gLmRfBzOnDKiLQXzCbP17sAPNdxQyUFltSRVThPyNsvVPCmNwNA4PGGk0KlyLIOGqPDs2x8drqnnG/zgKuWSSvvZhkY2Ne1HsINGegIOAHm+iJgRm03fahjXZdNyKtfW5jAQBjsIJr26dtsjdPi4JjJNhwGmrr" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=0,\\:\\mathrm{Horizontal}:\\:y=3" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{3x^{2}-18x+24}{x^{2}-4x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }