Factor and explain each step
2x^2-13x-45
factors of 2 -->1, 2
factors of 45 -->1 ,9,5,3,3,15 (1,45)(9,5)..etc
well 2 *9 is 18 and 18-5 = 13 so try them
(2x + 5)(x-9)
First you need to multiply the first and last digits in the equation (2 and 45). Now you find factors of 90 that will equal 13 when added. Both 18 and 5 go into 90 and when used appropriately, they add to -13. So to write it in its new form it would look like 2x^2-18x+5x-45. Now put parantheses around 2x^2-18x and another set around 5x-45. Notice that in the first set of parantheses that 2 and x are in both numbers. And in the second the number 5 is in both. So it will now look like 2x(x-9)+5(x-9). The function within the parantheses will be the same every time (after you factor the two set of numbers of course). Now you take what is in the parantheses and use that as part of the answer. You also use the part outside of the parantheses as well. So your answer should look like (2x+5)(x-9). The two parantheses can be switched.
To factor the quadratic expression 2x^2 - 13x - 45, we can use a method called "factoring by grouping" or "splitting the middle term."
Step 1: Multiply the coefficient of x^2 (2) by the constant term (-45). In this case, 2 * -45 = -90.
Step 2: Find two numbers that multiply to -90 and add up to the coefficient of the x term (-13). In this case, the numbers are -18 and +5, because -18 * 5 = -90 and -18 + 5 = -13.
Step 3: Rewrite the middle term (-13x) using the two numbers found in Step 2. The expression becomes 2x^2 - 18x + 5x - 45.
Step 4: Group the terms using parentheses: (2x^2 - 18x) + (5x - 45).
Step 5: Factor out the greatest common factor (GCF) from each group. In the first group, the GCF is 2x, and in the second group, the GCF is 5. Factoring out the GCF, we get 2x(x - 9) + 5(x - 9).
Step 6: Notice that both groups now have a common factor, which is (x - 9). Factor this common binomial out of the expression: (2x + 5)(x - 9).
Hence, we have factored the quadratic expression 2x^2 - 13x - 45 as (2x + 5)(x - 9).