factor x^2-x-20

{ "query": { "display": "factor $$x^{2}-x-20$$", "symbolab_question": "FACTOR#factor x^{2}-x-20" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(x+4)(x-5)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$x^{2}-x-20:{\\quad}\\left(x+4\\right)\\left(x-5\\right)$$", "input": "x^{2}-x-20", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-x-20", "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": [ "$$a=1,\\:b=-1,\\:c=-20$$", "$$u*v=-20,\\:u+v=-1$$" ] }, { "type": "interim", "title": "Factors of $$20:{\\quad}1,\\:2,\\:4,\\:5,\\:10,\\:20$$", "input": "20", "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 $$20:{\\quad}2,\\:2,\\:5$$", "input": "20", "steps": [ { "type": "step", "primary": "$$20\\:$$divides by $$2\\quad\\:20=10\\cdot\\:2$$", "result": "=2\\cdot\\:10" }, { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpfAIYH0B5pAUto4RYxQ3UuBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKptlmTzO+oYqKr2Nj1dLY7LNiunDvEaxn8n7nBifSeyk" } }, { "type": "interim", "title": "Multiply the prime factors of $$20:{\\quad}4,\\:10$$", "result": "4,\\:10", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:5=10$$" ] }, { "type": "step", "result": "4,\\:10" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$20\\:$$ itself", "result": "1,\\:20" }, { "type": "step", "primary": "The factors of $$20$$", "result": "1,\\:2,\\:4,\\:5,\\:10,\\:20" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$20:{\\quad}-1,\\:-2,\\:-4,\\:-5,\\:-10,\\:-20$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-5,\\:-10,\\:-20" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-20,\\:$$check if $$u+v=-1$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-20:\\quad\\:u*v=-20,\\:u+v=-19\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=-10:\\quad\\:u*v=-20,\\:u+v=-8\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=4,\\:v=-5:\\quad\\:u*v=-20,\\:u+v=-1\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=5,\\:v=-4:\\quad\\:u*v=-20,\\:u+v=1\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=10,\\:v=-2:\\quad\\:u*v=-20,\\:u+v=8\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=20,\\:v=-1:\\quad\\:u*v=-20,\\:u+v=19\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=4,\\:v=-5" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}+4x\\right)+\\left(-5x-20\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwiTAtHwe77KW105uEUM7Z4osjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKE8+48ptyTmxmUSr1Die6IXWkk+Kt1eMHZLTaqpS1jGzZrqMyF3nXkDK7FLxfFx124Zi9wpyEKv79aqbxM02VJiJ4DPJZQ/Gd4qNOAL0ITrMb8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "result": "=\\left(x^{2}+4x\\right)+\\left(-5x-20\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}+4x:\\quad\\:x\\left(x+4\\right)$$", "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": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fyAweyQvVwJsSQQKDjm97PUMABIQKMSMaYSzBXqqWZF1C4k27jzuj6KLnWyn2hb8bFwMwSf2fjKgNqIbLArbNOLWXfeXo8KExseWXOwkmyd24affICWdjh78fVRVNyyCmi2FsF7QWDlTFS2ckOasv2DsCfsZRv7rGNL0ROttohawiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Factor out $$-5\\:$$from $$-5x-20:\\quad\\:-5\\left(x+4\\right)$$", "input": "-5x-20", "steps": [ { "type": "step", "primary": "Rewrite $$-20$$ as $$-5\\cdot\\:4$$", "result": "=-5x-5\\cdot\\:4" }, { "type": "step", "primary": "Factor out common term $$-5$$", "result": "=-5\\left(x+4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7azZp4DB+B1N0dq43DKjaOxyOUtErWNRXbTqqjj69E3uBgn/GSIMdAWH6UHJYQzHa0KOEq32rRoLFZaty3uWqn0V5htgN+qCwR6RjLiy0n4bMPF9tfONRiFlp3oxDMg6VTbwiB3igy080xFttLaAynVo1nK6/qb8le5s1ntZmdqskt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=x\\left(x+4\\right)-5\\left(x+4\\right)" }, { "type": "step", "primary": "Factor out common term $$x+4$$", "result": "=\\left(x+4\\right)\\left(x-5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }