factor 4k^2-17k+4

{ "query": { "display": "factor $$4k^{2}-17k+4$$", "symbolab_question": "FACTOR#factor 4k^{2}-17k+4" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(4k-1)(k-4)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$4k^{2}-17k+4:{\\quad}\\left(4k-1\\right)\\left(k-4\\right)$$", "input": "4k^{2}-17k+4", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "4k^{2}-17k+4", "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": [ "$$a=4,\\:b=-17,\\:c=4$$", "$$u*v=16,\\:u+v=-17$$" ] }, { "type": "interim", "title": "Factors of $$16:{\\quad}1,\\:2,\\:4,\\:8,\\:16$$", "input": "16", "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 $$16:{\\quad}2,\\:2,\\:2,\\:2$$", "input": "16", "steps": [ { "type": "step", "primary": "$$16\\:$$divides by $$2\\quad\\:16=8\\cdot\\:2$$", "result": "=2\\cdot\\:8" }, { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\: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\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsgwKbhkhYm60j+YKcxMBSGBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGVsI5B83GgK4Rq+y4sowjuo" } }, { "type": "interim", "title": "Multiply the prime factors of $$16:{\\quad}4,\\:8$$", "result": "4,\\:8", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:2\\cdot\\:2=8$$" ] }, { "type": "step", "result": "4,\\:8" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$16\\:$$ itself", "result": "1,\\:16" }, { "type": "step", "primary": "The factors of $$16$$", "result": "1,\\:2,\\:4,\\:8,\\:16" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$16:{\\quad}-1,\\:-2,\\:-4,\\:-8,\\:-16$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-8,\\:-16" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=16,\\:$$check if $$u+v=-17$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=16:\\quad\\:u*v=16,\\:u+v=17\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=8:\\quad\\:u*v=16,\\:u+v=10\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=4,\\:v=4:\\quad\\:u*v=16,\\:u+v=8\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-16:\\quad\\:u*v=16,\\:u+v=-17\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=-2,\\:v=-8:\\quad\\:u*v=16,\\:u+v=-10\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-4,\\:v=-4:\\quad\\:u*v=16,\\:u+v=-8\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-16" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(4k^{2}-k\\right)+\\left(-16k+4\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwhglMrAY7Hfba2bEVYkHRFRJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMNnkMpaZsYuvGk1zw9xSkBOm8l8zUGxXHyIVnWNZEmi6k3kCh3oevUunZ7/b0qFKBSWsWV28wexhpgboh1m5Hi0Fimz7IJvR01OX8go+hj05ompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\left(4k^{2}-k\\right)+\\left(-16k+4\\right)" }, { "type": "interim", "title": "Factor out $$k\\:$$from $$4k^{2}-k:\\quad\\:k\\left(4k-1\\right)$$", "input": "4k^{2}-k", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$k^{2}=kk$$" ], "result": "=4kk-k", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$k$$", "result": "=k\\left(4k-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75g1BiNojIUS69UNUcirKBY7LYm+71IsB3xNsnjnlFFF1C4k27jzuj6KLnWyn2hb8aXvCNLbt4VnFjdTsvDjEfb6B3wRBkftMXxzEhH09cO6F4aOVLaBuuVKS9lqM99wwcajz1dNoYJeUot2MGkatvhVbFv5tn3SzAUILtN8f5m016UVDN3+r3twFG3wm1k1k" } }, { "type": "interim", "title": "Factor out $$-4\\:$$from $$-16k+4:\\quad\\:-4\\left(4k-1\\right)$$", "input": "-16k+4", "steps": [ { "type": "step", "primary": "Rewrite $$-16$$ as $$-4\\cdot\\:4$$", "secondary": [ "Rewrite $$4$$ as $$-4\\cdot\\:1$$" ], "result": "=-4\\cdot\\:4k+4\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$-4$$", "result": "=-4\\left(4k-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TSpD/NRtIKMujVf1J/0aihyOUtErWNRXbTqqjj69E3uBgn/GSIMdAWH6UHJYQzHazhrGyFBawDWA75SPOf+T2uiGloKKN9A2push0xMbRtt24affICWdjh78fVRVNyyCuCVo5MXxyJSeMHBYGgHWoV47LAVx32n7Uuw0kzRhj/S/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=k\\left(4k-1\\right)-4\\left(4k-1\\right)" }, { "type": "step", "primary": "Factor out common term $$4k-1$$", "result": "=\\left(4k-1\\right)\\left(k-4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }