factor-x^2-5x+24

{ "query": { "display": "factor $$-x^{2}-5x+24$$", "symbolab_question": "FACTOR#factor -x^{2}-5x+24" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "-(x-3)(x+8)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$-x^{2}-5x+24:{\\quad}-\\left(x-3\\right)\\left(x+8\\right)$$", "input": "-x^{2}-5x+24", "steps": [ { "type": "interim", "title": "Factor out $$-1:{\\quad}-\\left(x^{2}+5x-24\\right)$$", "input": "-x^{2}-5x+24", "steps": [ { "type": "step", "primary": "Rewrite $$24$$ as $$-\\left(-24\\right)$$", "result": "=-x^{2}-5x-\\left(-24\\right)" }, { "type": "step", "primary": "Factor out $$-1$$", "result": "=-\\left(x^{2}+5x-24\\right)" } ], "meta": { "solvingClass": "Solver2", "interimType": "Factor Out Specific 1Eq" } }, { "type": "step", "result": "=-\\left(x^{2}+5x-24\\right)" }, { "type": "interim", "title": "Factor $$x^{2}+5x-24:{\\quad}\\left(x-3\\right)\\left(x+8\\right)$$", "input": "x^{2}+5x-24", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+5x-24", "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": [ "$$a=1,\\:b=5,\\:c=-24$$", "$$u*v=-24,\\:u+v=5$$" ] }, { "type": "interim", "title": "Factors of $$24:{\\quad}1,\\:2,\\:3,\\:4,\\:6,\\:8,\\:12,\\:24$$", "input": "24", "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 $$24:{\\quad}2,\\:2,\\:2,\\:3$$", "input": "24", "steps": [ { "type": "step", "primary": "$$24\\:$$divides by $$2\\quad\\:24=12\\cdot\\:2$$", "result": "=2\\cdot\\:12" }, { "type": "step", "primary": "$$12\\:$$divides by $$2\\quad\\:12=6\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:6" }, { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuyvdYdg8xJlsOw+n4Rzo2GBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKptlmTzO+oYqKr2Nj1dLY7Lk/uIB5yXjV0QE57O2LF0+" } }, { "type": "interim", "title": "Multiply the prime factors of $$24:{\\quad}4,\\:8,\\:6,\\:12$$", "result": "4,\\:8,\\:6,\\:12", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:2\\cdot\\:2=8$$", "$$2\\cdot\\:3=6$$", "$$2\\cdot\\:2\\cdot\\:3=12$$" ] }, { "type": "step", "result": "4,\\:8,\\:6,\\:12" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:3" }, { "type": "step", "primary": "Add 1 and the number $$24\\:$$ itself", "result": "1,\\:24" }, { "type": "step", "primary": "The factors of $$24$$", "result": "1,\\:2,\\:3,\\:4,\\:6,\\:8,\\:12,\\:24" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$24:{\\quad}-1,\\:-2,\\:-3,\\:-4,\\:-6,\\:-8,\\:-12,\\:-24$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-3,\\:-4,\\:-6,\\:-8,\\:-12,\\:-24" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-24,\\:$$check if $$u+v=5$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-24:\\quad\\:u*v=-24,\\:u+v=-23\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=-12:\\quad\\:u*v=-24,\\:u+v=-10\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=3,\\:v=-8:\\quad\\:u*v=-24,\\:u+v=-5\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=4,\\:v=-6:\\quad\\:u*v=-24,\\:u+v=-2\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=6,\\:v=-4:\\quad\\:u*v=-24,\\:u+v=2\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=8,\\:v=-3:\\quad\\:u*v=-24,\\:u+v=5\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=12,\\:v=-2:\\quad\\:u*v=-24,\\:u+v=10\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=24,\\:v=-1:\\quad\\:u*v=-24,\\:u+v=23\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=8,\\:v=-3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-3x\\right)+\\left(8x-24\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwjGmeYyS4tCg9A87QRMcVZErnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI/r6gi+Yih/GCZiW1Dagm72A7XO2XK/mxf02VoyMUquCprqMyF3nXkDK7FLxfFx124Zi9wpyEKv79aqbxM02VJiJ4DPJZQ/Gd4qNOAL0ITrMb8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "result": "=\\left(x^{2}-3x\\right)+\\left(8x-24\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-3x:\\quad\\:x\\left(x-3\\right)$$", "input": "x^{2}-3x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-3x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IGfsPE9RU3/JW76ghU+JwDAclYiY05CGstcAePg+jbh1C4k27jzuj6KLnWyn2hb8bFwMwSf2fjKgNqIbLArbNLjTbNB6I4gQBdwlnqbCldt24affICWdjh78fVRVNyyCmi2FsF7QWDlTFS2ckOasv8NOHvTBuRgHCvEz5j2vstqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Factor out $$8\\:$$from $$8x-24:\\quad\\:8\\left(x-3\\right)$$", "input": "8x-24", "steps": [ { "type": "step", "primary": "Rewrite $$24$$ as $$8\\cdot\\:3$$", "result": "=8x-8\\cdot\\:3" }, { "type": "step", "primary": "Factor out common term $$8$$", "result": "=8\\left(x-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ONK63G4gw+5N6/hm8MKvUZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwgH5KzeZTgQ4sBnlx8lPQb2RLd2VwIqlBNByF6663sySuUTSpB0am5HPeyxkjakcLB289thNOM7FPdKKDT/dY58vywE3AMd62SYFYI74g1Rg=" } }, { "type": "step", "result": "=x\\left(x-3\\right)+8\\left(x-3\\right)" }, { "type": "step", "primary": "Factor out common term $$x-3$$", "result": "=\\left(x-3\\right)\\left(x+8\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "=-\\left(x-3\\right)\\left(x+8\\right)" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }