Can you show me how to factor 2x^2+11x+5 with the X method?
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To factor the quadratic expression 2x squared plus 11x plus 5 using the X method, we first identify the coefficients. In the standard form ax squared plus bx plus c, we have a equals 2, b equals 11, and c equals 5. The X method helps us find two numbers that multiply to give a times c, which is 10, and add up to b, which is 11. We'll set up our X with 10 at the top and 11 at the bottom.
Now we need to find two numbers that multiply to give 10 and add up to 11. Let's try some factors of 10. The factors of 10 are 1 and 10, or 2 and 5. If we check 1 and 10, we get 1 times 10 equals 10, and 1 plus 10 equals 11. Perfect! These are the numbers we need. We'll place 1 and 10 on the X to show they're our factors.
Now that we've found our two numbers, 1 and 10, we'll use them to split the middle term 11x. Since 1 plus 10 equals 11, we can rewrite 11x as 1x plus 10x. So our expression becomes 2x squared plus 1x plus 10x plus 5. Next, we'll group these terms into two pairs: 2x squared plus 1x in the first group, and 10x plus 5 in the second group. This grouping will help us identify common factors in the next step.
Now we'll factor out the greatest common factor from each group. From the first group, 2x squared plus 1x, we can factor out x to get x times 2x plus 1. From the second group, 10x plus 5, we can factor out 5 to get 5 times 2x plus 1. Notice that both groups now have the common factor of 2x plus 1. We can factor this out to get 2x plus 1 times x plus 5. This is our final factored form of the original expression.
Let's summarize the X method for factoring quadratics. We started with the expression 2x squared plus 11x plus 5. We identified that a equals 2, b equals 11, and c equals 5, giving us a times c equals 10. We found that 1 and 10 are the two numbers that multiply to 10 and add to 11. We split the middle term 11x into 1x plus 10x, giving us 2x squared plus 1x plus 10x plus 5. We grouped the terms and factored out common factors: x times 2x plus 1, plus 5 times 2x plus 1. Finally, we factored out the common binomial 2x plus 1 to get our answer: 2x plus 1 times x plus 5. This is the factored form of our original quadratic expression.