factor 8y^2+30y+7

{ "query": { "display": "factor $$8y^{2}+30y+7$$", "symbolab_question": "FACTOR#factor 8y^{2}+30y+7" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(4y+1)(2y+7)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$8y^{2}+30y+7:{\\quad}\\left(4y+1\\right)\\left(2y+7\\right)$$", "input": "8y^{2}+30y+7", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "8y^{2}+30y+7", "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": [ "$$a=8,\\:b=30,\\:c=7$$", "$$u*v=56,\\:u+v=30$$" ] }, { "type": "interim", "title": "Factors of $$56:{\\quad}1,\\:2,\\:4,\\:7,\\:8,\\:14,\\:28,\\:56$$", "input": "56", "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 $$56:{\\quad}2,\\:2,\\:2,\\:7$$", "input": "56", "steps": [ { "type": "step", "primary": "$$56\\:$$divides by $$2\\quad\\:56=28\\cdot\\:2$$", "result": "=2\\cdot\\:28" }, { "type": "step", "primary": "$$28\\:$$divides by $$2\\quad\\:28=14\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:14" }, { "type": "step", "primary": "$$14\\:$$divides by $$2\\quad\\:14=7\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:7" }, { "type": "step", "primary": "$$2,\\:7$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRrQXvWW1cWhvoVZffAAUVfOBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKg2S/x/kiCP8yWoLVdRKyxhsI5B83GgK4Rq+y4sowjuo" } }, { "type": "interim", "title": "Multiply the prime factors of $$56:{\\quad}4,\\:8,\\:14,\\:28$$", "result": "4,\\:8,\\:14,\\:28", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:2\\cdot\\:2=8$$", "$$2\\cdot\\:7=14$$", "$$2\\cdot\\:2\\cdot\\:7=28$$" ] }, { "type": "step", "result": "4,\\:8,\\:14,\\:28" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:7" }, { "type": "step", "primary": "Add 1 and the number $$56\\:$$ itself", "result": "1,\\:56" }, { "type": "step", "primary": "The factors of $$56$$", "result": "1,\\:2,\\:4,\\:7,\\:8,\\:14,\\:28,\\:56" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=56,\\:$$check if $$u+v=30$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=56:\\quad\\:u*v=56,\\:u+v=57\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=28:\\quad\\:u*v=56,\\:u+v=30\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=4,\\:v=14:\\quad\\:u*v=56,\\:u+v=18\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=7,\\:v=8:\\quad\\:u*v=56,\\:u+v=15\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=2,\\:v=28" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(8y^{2}+2y\\right)+\\left(28y+7\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwpRA/kkkw0yIiP0DQYZZ/LVJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMPKE2o7vQKuJC7nc6HzFFj5FfW7I4niLrOW39t4RmwVZk3kCh3oevUunZ7/b0qFKBSWsWV28wexhpgboh1m5Hi0Fimz7IJvR01OX8go+hj05ompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\left(8y^{2}+2y\\right)+\\left(28y+7\\right)" }, { "type": "interim", "title": "Factor out $$2y\\:$$from $$8y^{2}+2y:\\quad\\:2y\\left(4y+1\\right)$$", "input": "8y^{2}+2y", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{2}=yy$$" ], "result": "=8yy+2y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite $$8$$ as $$2\\cdot\\:4$$", "secondary": [ "Rewrite $$2$$ as $$2\\cdot\\:1$$" ], "result": "=2\\cdot\\:4yy+2\\cdot\\:1\\cdot\\:y" }, { "type": "step", "primary": "Factor out common term $$2y$$", "result": "=2y\\left(4y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XhbtWYelNs0T39onWWS8pjykD1Zu/8AtylhRc3WLA6wrD5aUE2ezECPCoH/ZdaT9sggGzv7GXRxwLPFlb8Dss9j7xkKqYoJ5+FA+6bGcPCF2YTPJQsJxQm02Mm9NMgv53ZTzUrgOcUjEf+k6IwL09b90gYh+voem1Ry/C7WYvZshAP0KGTkDhPKwKIVBL1mk" } }, { "type": "interim", "title": "Factor out $$7\\:$$from $$28y+7:\\quad\\:7\\left(4y+1\\right)$$", "input": "28y+7", "steps": [ { "type": "step", "primary": "Rewrite $$28$$ as $$7\\cdot\\:4$$", "secondary": [ "Rewrite $$7$$ as $$7\\cdot\\:1$$" ], "result": "=7\\cdot\\:4y+7\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$7$$", "result": "=7\\left(4y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72UBR4JnNmRhKNMN+Kz+XSJN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwN74QrTbw7PjY3Aa7hVlWCiGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDuxBst4YRa/Yxo+odDVxZUiqz9Q6x2hrAcKFg3xF+gGRFw=" } }, { "type": "step", "result": "=2y\\left(4y+1\\right)+7\\left(4y+1\\right)" }, { "type": "step", "primary": "Factor out common term $$4y+1$$", "result": "=\\left(4y+1\\right)\\left(2y+7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }