factor 5k^2-18k+9

{ "query": { "display": "factor $$5k^{2}-18k+9$$", "symbolab_question": "FACTOR#factor 5k^{2}-18k+9" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(5k-3)(k-3)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$5k^{2}-18k+9:{\\quad}\\left(5k-3\\right)\\left(k-3\\right)$$", "input": "5k^{2}-18k+9", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "5k^{2}-18k+9", "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": [ "$$a=5,\\:b=-18,\\:c=9$$", "$$u*v=45,\\:u+v=-18$$" ] }, { "type": "interim", "title": "Factors of $$45:{\\quad}1,\\:3,\\:5,\\:9,\\:15,\\:45$$", "input": "45", "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 $$45:{\\quad}3,\\:3,\\:5$$", "input": "45", "steps": [ { "type": "step", "primary": "$$45\\:$$divides by $$3\\quad\\:45=15\\cdot\\:3$$", "result": "=3\\cdot\\:15" }, { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "$$3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=3\\cdot\\:3\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRghzcFN3K5+bODbQgDGO37eBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKuZYRbDpkVqH5Klb5a1uriWfcjVNh1BqSaCrmPN07Wzw" } }, { "type": "interim", "title": "Multiply the prime factors of $$45:{\\quad}9,\\:15$$", "result": "9,\\:15", "steps": [ { "type": "step", "primary": "$$3\\cdot\\:3=9$$", "secondary": [ "$$3\\cdot\\:5=15$$" ] }, { "type": "step", "result": "9,\\:15" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "3,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$45\\:$$ itself", "result": "1,\\:45" }, { "type": "step", "primary": "The factors of $$45$$", "result": "1,\\:3,\\:5,\\:9,\\:15,\\:45" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$45:{\\quad}-1,\\:-3,\\:-5,\\:-9,\\:-15,\\:-45$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-3,\\:-5,\\:-9,\\:-15,\\:-45" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=45,\\:$$check if $$u+v=-18$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=45:\\quad\\:u*v=45,\\:u+v=46\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=3,\\:v=15:\\quad\\:u*v=45,\\:u+v=18\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=9:\\quad\\:u*v=45,\\:u+v=14\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-45:\\quad\\:u*v=45,\\:u+v=-46\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-3,\\:v=-15:\\quad\\:u*v=45,\\:u+v=-18\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=-5,\\:v=-9:\\quad\\:u*v=45,\\:u+v=-14\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-3,\\:v=-15" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(5k^{2}-3k\\right)+\\left(-15k+9\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwj2wDlKYeRTdoD8bt6w1RMVJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMPLgE04JWX3ESYOYWlWWLbN1FR7jPEm1G7XqWNQMDgLr3ql8XXPq6bNQlMm+36iNhn60BI812H7zLg4XCKXFe8BJ8d96ThpuB4psnVRIBEwSb1FBr5UXMxw7y43bLdm5aQ=" } }, { "type": "step", "result": "=\\left(5k^{2}-3k\\right)+\\left(-15k+9\\right)" }, { "type": "interim", "title": "Factor out $$k\\:$$from $$5k^{2}-3k:\\quad\\:k\\left(5k-3\\right)$$", "input": "5k^{2}-3k", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$k^{2}=kk$$" ], "result": "=5kk-3k", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$k$$", "result": "=k\\left(5k-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YgXzQa0Bh+6YvyG+nfEFz2FE2yU4Nt/AemuIxmSG46IrD5aUE2ezECPCoH/ZdaT9daUTqZMrrNms8vZtXQT0gbPP3NMH4VE63cQh97tbfMa/Yn8ofzwCa9Wi6aIKxDLz28GN9vm9Fa155gOIWrCeTEqzC5zywdh4ZyBexpzx+w95hYZb4n3suB0LqCR0J9L8" } }, { "type": "interim", "title": "Factor out $$-3\\:$$from $$-15k+9:\\quad\\:-3\\left(5k-3\\right)$$", "input": "-15k+9", "steps": [ { "type": "step", "primary": "Rewrite $$-15$$ as $$-3\\cdot\\:5$$", "secondary": [ "Rewrite $$9$$ as $$-3\\cdot\\:3$$" ], "result": "=-3\\cdot\\:5k+3\\cdot\\:3" }, { "type": "step", "primary": "Factor out common term $$-3$$", "result": "=-3\\left(5k-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7c2fe5TEtR9R+6I/C+q/+3xyOUtErWNRXbTqqjj69E3uBgn/GSIMdAWH6UHJYQzHaEKzKcMboXYyeM/bZuz5BdbjTbNB6I4gQBdwlnqbCldt24affICWdjh78fVRVNyyCjRhUKumW11JgP7P3Q4729z3czyG8tUFIk1rD90oZYDy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=k\\left(5k-3\\right)-3\\left(5k-3\\right)" }, { "type": "step", "primary": "Factor out common term $$5k-3$$", "result": "=\\left(5k-3\\right)\\left(k-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }