factor 5x^2+2x-3

{ "query": { "display": "factor $$5x^{2}+2x-3$$", "symbolab_question": "FACTOR#factor 5x^{2}+2x-3" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(5x-3)(x+1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$5x^{2}+2x-3:{\\quad}\\left(5x-3\\right)\\left(x+1\\right)$$", "input": "5x^{2}+2x-3", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "5x^{2}+2x-3", "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": [ "$$a=5,\\:b=2,\\:c=-3$$", "$$u*v=-15,\\:u+v=2$$" ] }, { "type": "interim", "title": "Factors of $$15:{\\quad}1,\\:3,\\:5,\\:15$$", "input": "15", "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 $$15:{\\quad}3,\\:5$$", "input": "15", "steps": [ { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=3\\cdot\\:5" }, { "type": "step", "primary": "$$3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=3\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRgP6y8qhOrbDVDXhn1mWL/uBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGWfcjVNh1BqSaCrmPN07Wzw" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "3,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$15\\:$$ itself", "result": "1,\\:15" }, { "type": "step", "primary": "The factors of $$15$$", "result": "1,\\:3,\\:5,\\:15" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$15:{\\quad}-1,\\:-3,\\:-5,\\:-15$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-3,\\:-5,\\:-15" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-15,\\:$$check if $$u+v=2$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-15:\\quad\\:u*v=-15,\\:u+v=-14\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=3,\\:v=-5:\\quad\\:u*v=-15,\\:u+v=-2\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=-3:\\quad\\:u*v=-15,\\:u+v=2\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=15,\\:v=-1:\\quad\\:u*v=-15,\\:u+v=14\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=5,\\:v=-3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(5x^{2}-3x\\right)+\\left(5x-3\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwita2qdpEP9u4Lm9HtLT58ornvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI8ZdcayHqpsx2SVs4myXXqGYnY2UX/AsOnmXf5GBX8+FprqMyF3nXkDK7FLxfFx124Zi9wpyEKv79aqbxM02VJiJ4DPJZQ/Gd4qNOAL0ITrMb8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "result": "=\\left(5x^{2}-3x\\right)+\\left(5x-3\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$5x^{2}-3x:\\quad\\:x\\left(5x-3\\right)$$", "input": "5x^{2}-3x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=5xx-3x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(5x-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74pnWDu/HeHOrwj4GlNiWdYAyyKK2c4C139aOq8tDVLwrD5aUE2ezECPCoH/ZdaT9njZY7CyXWv0oHaGDmjrugP3caeS08M2FzIj/iFNyr76/Yn8ofzwCa9Wi6aIKxDLzUvWDKQVeo+SH+K3P55a4xmpiEPBMHaT24KHaVdI5aLpeWIHYZVfTMlyIRNexbDNd" } }, { "type": "step", "result": "=x\\left(5x-3\\right)+\\left(5x-3\\right)" }, { "type": "interim", "title": "Factor out common term $$\\left(5x-3\\right):{\\quad}\\left(5x-3\\right)\\left(x+1\\right)$$", "input": "x\\left(5x-3\\right)+\\left(5x-3\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\left(5x-3\\right)$$ as $$1\\cdot\\:\\left(5x-3\\right)$$", "result": "=x\\left(5x-3\\right)+1\\cdot\\:\\left(5x-3\\right)" }, { "type": "step", "primary": "Factor out common term $$5x-3$$", "result": "=\\left(5x-3\\right)\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Factor Take Out Common Term Title 1Eq" } }, { "type": "step", "result": "=\\left(5x-3\\right)\\left(x+1\\right)" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }