factor 2y^2+11y+15

{ "query": { "display": "factor $$2y^{2}+11y+15$$", "symbolab_question": "FACTOR#factor 2y^{2}+11y+15" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(2y+5)(y+3)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$2y^{2}+11y+15:{\\quad}\\left(2y+5\\right)\\left(y+3\\right)$$", "input": "2y^{2}+11y+15", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2y^{2}+11y+15", "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": [ "$$a=2,\\:b=11,\\:c=15$$", "$$u*v=30,\\:u+v=11$$" ] }, { "type": "interim", "title": "Factors of $$30:{\\quad}1,\\:2,\\:3,\\:5,\\:6,\\:10,\\:15,\\:30$$", "input": "30", "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 $$30:{\\quad}2,\\:3,\\:5$$", "input": "30", "steps": [ { "type": "step", "primary": "$$30\\:$$divides by $$2\\quad\\:30=15\\cdot\\:2$$", "result": "=2\\cdot\\:15" }, { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=2\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRoAjF7h2GnIiP5El3Wr6ML2BBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKhuQbtB5yLYoe1JMepWppwPNiunDvEaxn8n7nBifSeyk" } }, { "type": "interim", "title": "Multiply the prime factors of $$30:{\\quad}6,\\:10,\\:15$$", "result": "6,\\:10,\\:15", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:3=6$$", "secondary": [ "$$2\\cdot\\:5=10$$", "$$3\\cdot\\:5=15$$" ] }, { "type": "step", "result": "6,\\:10,\\:15" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:3,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$30\\:$$ itself", "result": "1,\\:30" }, { "type": "step", "primary": "The factors of $$30$$", "result": "1,\\:2,\\:3,\\:5,\\:6,\\:10,\\:15,\\:30" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=30,\\:$$check if $$u+v=11$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=30:\\quad\\:u*v=30,\\:u+v=31\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=15:\\quad\\:u*v=30,\\:u+v=17\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=3,\\:v=10:\\quad\\:u*v=30,\\:u+v=13\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=6:\\quad\\:u*v=30,\\:u+v=11\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=5,\\:v=6" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2y^{2}+5y\\right)+\\left(6y+15\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwhfUf8Jt0JWqx2Wv79VaYNmk3hxk9aCfAWodBRxXgUexnyLfTdVp2W4I+K/E5MHE+hcj6isJZHmVLbwm+mZ9aY4RXPaGMRZXdjcHoDP0iytRo3FrbIVCfwtwFPaso4bCl3ql8XXPq6bNQlMm+36iNhn60BI812H7zLg4XCKXFe8BJ8d96ThpuB4psnVRIBEwSb1FBr5UXMxw7y43bLdm5aQ=" } }, { "type": "step", "result": "=\\left(2y^{2}+5y\\right)+\\left(6y+15\\right)" }, { "type": "interim", "title": "Factor out $$y\\:$$from $$2y^{2}+5y:\\quad\\:y\\left(2y+5\\right)$$", "input": "2y^{2}+5y", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{2}=yy$$" ], "result": "=2yy+5y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(2y+5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7naIlrcNPBUMns2vVwwdwtKW0KLT+h4+1u9xtZNwi7nQrD5aUE2ezECPCoH/ZdaT9T9SuE8yQN9zI3YkPpnh9dR5CxsEAOFXIGVvGBU8VchC/Yn8ofzwCa9Wi6aIKxDLzMzo5Dlv5vW+wu0NcSR6fWN1Pkp/7vGkhdVJ7OOS7jZFhChU5drw4cW1oDmvhHS1m" } }, { "type": "interim", "title": "Factor out $$3\\:$$from $$6y+15:\\quad\\:3\\left(2y+5\\right)$$", "input": "6y+15", "steps": [ { "type": "step", "primary": "Rewrite $$6$$ as $$3\\cdot\\:2$$", "secondary": [ "Rewrite $$15$$ as $$3\\cdot\\:5$$" ], "result": "=3\\cdot\\:2y+3\\cdot\\:5" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(2y+5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oHNZ+k2+L82er4+PPmd3AZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwzc0TVfAeXdklYbmH7IgrBCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDux+yo2ZrvZn74MLNd65Gl+PGL2Mcb/vgkwkrftr0/ZC3c=" } }, { "type": "step", "result": "=y\\left(2y+5\\right)+3\\left(2y+5\\right)" }, { "type": "step", "primary": "Factor out common term $$2y+5$$", "result": "=\\left(2y+5\\right)\\left(y+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }