factor k^2-13k+22

{ "query": { "display": "factor $$k^{2}-13k+22$$", "symbolab_question": "FACTOR#factor k^{2}-13k+22" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(k-2)(k-11)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$k^{2}-13k+22:{\\quad}\\left(k-2\\right)\\left(k-11\\right)$$", "input": "k^{2}-13k+22", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "k^{2}-13k+22", "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": [ "$$a=1,\\:b=-13,\\:c=22$$", "$$u*v=22,\\:u+v=-13$$" ] }, { "type": "interim", "title": "Factors of $$22:{\\quad}1,\\:2,\\:11,\\:22$$", "input": "22", "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 $$22:{\\quad}2,\\:11$$", "input": "22", "steps": [ { "type": "step", "primary": "$$22\\:$$divides by $$2\\quad\\:22=11\\cdot\\:2$$", "result": "=2\\cdot\\:11" }, { "type": "step", "primary": "$$2,\\:11$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:11" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpOt/T2V/DHCaVZMQVnxtbSBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKptlmTzO+oYqKr2Nj1dLY7J1jVo3NKtIGSIqrPx5gObx" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:11" }, { "type": "step", "primary": "Add 1 and the number $$22\\:$$ itself", "result": "1,\\:22" }, { "type": "step", "primary": "The factors of $$22$$", "result": "1,\\:2,\\:11,\\:22" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$22:{\\quad}-1,\\:-2,\\:-11,\\:-22$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-11,\\:-22" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=22,\\:$$check if $$u+v=-13$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=22:\\quad\\:u*v=22,\\:u+v=23\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=11:\\quad\\:u*v=22,\\:u+v=13\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-22:\\quad\\:u*v=22,\\:u+v=-23\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-2,\\:v=-11:\\quad\\:u*v=22,\\:u+v=-13\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-2,\\:v=-11" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(k^{2}-2k\\right)+\\left(-11k+22\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwrhoPyQWRpgmAjALXeeK8JxJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMOoE8oLeijIdDRlAKYG3KSHQSd7SwxMVXr41cxyCGVMr3ql8XXPq6bNQlMm+36iNhn60BI812H7zLg4XCKXFe8BJ8d96ThpuB4psnVRIBEwSb1FBr5UXMxw7y43bLdm5aQ=" } }, { "type": "step", "result": "=\\left(k^{2}-2k\\right)+\\left(-11k+22\\right)" }, { "type": "interim", "title": "Factor out $$k\\:$$from $$k^{2}-2k:\\quad\\:k\\left(k-2\\right)$$", "input": "k^{2}-2k", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$k^{2}=kk$$" ], "result": "=kk-2k", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$k$$", "result": "=k\\left(k-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s716fUMYNj6GT4IEl9dgPhtQOU3jzXsX9ARbzd/1gptI51C4k27jzuj6KLnWyn2hb8tqFG2PisVxo0g1cD3HGGkGUEOft0UXEiE8pGKMwYoah24affICWdjh78fVRVNyyCNdMygafdeBctNvHIXnHQ4EqEHxJuf6uUYtF/Ta2lkxOwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Factor out $$-11\\:$$from $$-11k+22:\\quad\\:-11\\left(k-2\\right)$$", "input": "-11k+22", "steps": [ { "type": "step", "primary": "Rewrite $$22$$ as $$-11\\cdot\\:2$$", "result": "=-11k+11\\cdot\\:2" }, { "type": "step", "primary": "Factor out common term $$-11$$", "result": "=-11\\left(k-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74D2vq3eOC4ZYutBx0f+7vu6WhAHlhVR5DrUCfvc0BZjFeNKI8a2kiIgBIHIyg9qrJaH1+uOeEGFHGQUb4kuA2KxHFJ5IzjrM/YgjvQ6LTQeF4aOVLaBuuVKS9lqM99wwt2zQhj/9CHJl0GVeiciMgIWGXTrfgIf9hItGHNJqmafjn3yOFImNC2i+UAjYwPM0" } }, { "type": "step", "result": "=k\\left(k-2\\right)-11\\left(k-2\\right)" }, { "type": "step", "primary": "Factor out common term $$k-2$$", "result": "=\\left(k-2\\right)\\left(k-11\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }