SOLUTION: Factor completely 4x^2+32x+60

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Question 109508: Factor completely
4x^2+32x+60
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


4%2Ax%5E2%2B32%2Ax%2B60 Start with the given expression.



4%28x%5E2%2B8x%2B15%29 Factor out the GCF 4.



Now let's try to factor the inner expression x%5E2%2B8x%2B15



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Looking at the expression x%5E2%2B8x%2B15, we can see that the first coefficient is 1, the second coefficient is 8, and the last term is 15.



Now multiply the first coefficient 1 by the last term 15 to get %281%29%2815%29=15.



Now the question is: what two whole numbers multiply to 15 (the previous product) and add to the second coefficient 8?



To find these two numbers, we need to list all of the factors of 15 (the previous product).



Factors of 15:

1,3,5,15

-1,-3,-5,-15



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



These factors pair up and multiply to 15.

1*15 = 15
3*5 = 15
(-1)*(-15) = 15
(-3)*(-5) = 15


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 8:



First NumberSecond NumberSum
1151+15=16
353+5=8
-1-15-1+(-15)=-16
-3-5-3+(-5)=-8




From the table, we can see that the two numbers 3 and 5 add to 8 (the middle coefficient).



So the two numbers 3 and 5 both multiply to 15 and add to 8



Now replace the middle term 8x with 3x%2B5x. Remember, 3 and 5 add to 8. So this shows us that 3x%2B5x=8x.



x%5E2%2Bhighlight%283x%2B5x%29%2B15 Replace the second term 8x with 3x%2B5x.



%28x%5E2%2B3x%29%2B%285x%2B15%29 Group the terms into two pairs.



x%28x%2B3%29%2B%285x%2B15%29 Factor out the GCF x from the first group.



x%28x%2B3%29%2B5%28x%2B3%29 Factor out 5 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%2B5%29%28x%2B3%29 Combine like terms. Or factor out the common term x%2B3



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So 4%28x%5E2%2B8x%2B15%29 then factors further to 4%28x%2B5%29%28x%2B3%29



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Answer:



So 4%2Ax%5E2%2B32%2Ax%2B60 completely factors to 4%28x%2B5%29%28x%2B3%29.



In other words, 4%2Ax%5E2%2B32%2Ax%2B60=4%28x%2B5%29%28x%2B3%29.



Note: you can check the answer by expanding 4%28x%2B5%29%28x%2B3%29 to get 4%2Ax%5E2%2B32%2Ax%2B60 or by graphing the original expression and the answer (the two graphs should be identical).