SOLUTION: Hi. Factor completely 88-19w+w^2

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Hi. Factor completely 88-19w+w^2      Log On


   

Question 461734: Hi.
Factor completely
88-19w+w^2
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

88-19w%2Bw%5E2....rearrange terms
w%5E2-19w%2B88

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2-19x%2B88, we can see that the first coefficient is 1, the second coefficient is -19, and the last term is 88.



Now multiply the first coefficient 1 by the last term 88 to get %281%29%2888%29=88.



Now the question is: what two whole numbers multiply to 88 (the previous product) and add to the second coefficient -19?



To find these two numbers, we need to list all of the factors of 88 (the previous product).



Factors of 88:

1,2,4,8,11,22,44,88

-1,-2,-4,-8,-11,-22,-44,-88



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 88.

1*88 = 88
2*44 = 88
4*22 = 88
8*11 = 88
(-1)*(-88) = 88
(-2)*(-44) = 88
(-4)*(-22) = 88
(-8)*(-11) = 88


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -19:



First NumberSecond NumberSum
1881+88=89
2442+44=46
4224+22=26
8118+11=19
-1-88-1+(-88)=-89
-2-44-2+(-44)=-46
-4-22-4+(-22)=-26
-8-11-8+(-11)=-19




From the table, we can see that the two numbers -8 and -11 add to -19 (the middle coefficient).



So the two numbers -8 and -11 both multiply to 88 and add to -19



Now replace the middle term -19x with -8x-11x. Remember, -8 and -11 add to -19. So this shows us that -8x-11x=-19x.



x%5E2%2Bhighlight%28-8x-11x%29%2B88 Replace the second term -19x with -8x-11x.



%28x%5E2-8x%29%2B%28-11x%2B88%29 Group the terms into two pairs.



x%28x-8%29%2B%28-11x%2B88%29 Factor out the GCF x from the first group.



x%28x-8%29-11%28x-8%29 Factor out 11 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28x-11%29%28x-8%29 Combine like terms. Or factor out the common term x-8



===============================================================



Answer:



So x%5E2-19%2Ax%2B88 factors to %28x-11%29%28x-8%29.



In other words, x%5E2-19%2Ax%2B88=%28x-11%29%28x-8%29.



Note: you can check the answer by expanding %28x-11%29%28x-8%29 to get x%5E2-19%2Ax%2B88 or by graphing the original expression and the answer (the two graphs should be identical).