SOLUTION: How would you factor 5x^2-42x+16 using the box method?

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Question 251553: How would you factor 5x^2-42x+16 using the box method?
Found 2 solutions by richwmiller, jim_thompson5910:
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
http://www.purplemath.com/modules/factquad3.htm

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm not sure what you mean by the box method, but here's one way to do it.



Looking at the expression 5x%5E2-42x%2B16, we can see that the first coefficient is 5, the second coefficient is -42, and the last term is 16.


Now multiply the first coefficient 5 by the last term 16 to get %285%29%2816%29=80.


Now the question is: what two whole numbers multiply to 80 (the previous product) and add to the second coefficient -42?


To find these two numbers, we need to list all of the factors of 80 (the previous product).


Factors of 80:
1,2,4,5,8,10,16,20,40,80
-1,-2,-4,-5,-8,-10,-16,-20,-40,-80


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


These factors pair up and multiply to 80.
1*80 = 80
2*40 = 80
4*20 = 80
5*16 = 80
8*10 = 80
(-1)*(-80) = 80
(-2)*(-40) = 80
(-4)*(-20) = 80
(-5)*(-16) = 80
(-8)*(-10) = 80

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -42:


First NumberSecond NumberSum
1801+80=81
2402+40=42
4204+20=24
5165+16=21
8108+10=18
-1-80-1+(-80)=-81
-2-40-2+(-40)=-42
-4-20-4+(-20)=-24
-5-16-5+(-16)=-21
-8-10-8+(-10)=-18



From the table, we can see that the two numbers -2 and -40 add to -42 (the middle coefficient).


So the two numbers -2 and -40 both multiply to 80 and add to -42


Now replace the middle term -42x with -2x-40x. Remember, -2 and -40 add to -42. So this shows us that -2x-40x=-42x.


5x%5E2%2Bhighlight%28-2x-40x%29%2B16 Replace the second term -42x with -2x-40x.


%285x%5E2-2x%29%2B%28-40x%2B16%29 Group the terms into two pairs.


x%285x-2%29%2B%28-40x%2B16%29 Factor out the GCF x from the first group.


x%285x-2%29-8%285x-2%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%285x-2%29 Combine like terms. Or factor out the common term 5x-2


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Answer:


So 5x%5E2-42x%2B16 factors to %28x-8%29%285x-2%29.


In other words, 5x%5E2-42x%2B16=%28x-8%29%285x-2%29.


Note: you can check the answer by expanding %28x-8%29%285x-2%29 to get 5x%5E2-42x%2B16 or by graphing the original expression and the answer (the two graphs should be identical).