SOLUTION: How do I factor this equation? 7x squared - 60x + 32 = 0

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Question 693203: How do I factor this equation? 7x squared - 60x + 32 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 7x%5E2-60x%2B32, we can see that the first coefficient is 7, the second coefficient is -60, and the last term is 32.


Now multiply the first coefficient 7 by the last term 32 to get %287%29%2832%29=224.


Now the question is: what two whole numbers multiply to 224 (the previous product) and add to the second coefficient -60?


To find these two numbers, we need to list all of the factors of 224 (the previous product).


Factors of 224:
1,2,4,7,8,14,16,28,32,56,112,224
-1,-2,-4,-7,-8,-14,-16,-28,-32,-56,-112,-224


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


These factors pair up and multiply to 224.
1*224 = 224
2*112 = 224
4*56 = 224
7*32 = 224
8*28 = 224
14*16 = 224
(-1)*(-224) = 224
(-2)*(-112) = 224
(-4)*(-56) = 224
(-7)*(-32) = 224
(-8)*(-28) = 224
(-14)*(-16) = 224

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -60:


First NumberSecond NumberSum
12241+224=225
21122+112=114
4564+56=60
7327+32=39
8288+28=36
141614+16=30
-1-224-1+(-224)=-225
-2-112-2+(-112)=-114
-4-56-4+(-56)=-60
-7-32-7+(-32)=-39
-8-28-8+(-28)=-36
-14-16-14+(-16)=-30



From the table, we can see that the two numbers -4 and -56 add to -60 (the middle coefficient).


So the two numbers -4 and -56 both multiply to 224 and add to -60


Now replace the middle term -60x with -4x-56x. Remember, -4 and -56 add to -60. So this shows us that -4x-56x=-60x.


7x%5E2%2Bhighlight%28-4x-56x%29%2B32 Replace the second term -60x with -4x-56x.


%287x%5E2-4x%29%2B%28-56x%2B32%29 Group the terms into two pairs.


x%287x-4%29%2B%28-56x%2B32%29 Factor out the GCF x from the first group.


x%287x-4%29-8%287x-4%29 Factor out 8 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-8%29%287x-4%29 Combine like terms. Or factor out the common term 7x-4


So 7x%5E2-60x%2B32 factors to %28x-8%29%287x-4%29.


In other words, 7x%5E2-60x%2B32=%28x-8%29%287x-4%29.


Note: you can check the answer by expanding %28x-8%29%287x-4%29 to get 7x%5E2-60x%2B32 or by graphing the original expression and the answer (the two graphs should be identical).