SOLUTION: Rewrite the middle term as the sum of two terms and then factor by grouping. x^2+4x-12

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Rewrite the middle term as the sum of two terms and then factor by grouping. x^2+4x-12      Log On


   

Question 103932: Rewrite the middle term as the sum of two terms and then factor by grouping.
x^2+4x-12
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)


Looking at the expression x%5E2%2B4x-12, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is -12.



Now multiply the first coefficient 1 by the last term -12 to get %281%29%28-12%29=-12.



Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 4?



To find these two numbers, we need to list all of the factors of -12 (the previous product).



Factors of -12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to -12.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:



First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1




From the table, we can see that the two numbers -2 and 6 add to 4 (the middle coefficient).



So the two numbers -2 and 6 both multiply to -12 and add to 4



Now replace the middle term 4x with -2x%2B6x. Remember, -2 and 6 add to 4. So this shows us that -2x%2B6x=4x.



x%5E2%2Bhighlight%28-2x%2B6x%29-12 Replace the second term 4x with -2x%2B6x.



%28x%5E2-2x%29%2B%286x-12%29 Group the terms into two pairs.



x%28x-2%29%2B%286x-12%29 Factor out the GCF x from the first group.



x%28x-2%29%2B6%28x-2%29 Factor out 6 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%2B6%29%28x-2%29 Combine like terms. Or factor out the common term x-2



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



Answer:



So x%5E2%2B4%2Ax-12 factors to %28x%2B6%29%28x-2%29.



In other words, x%5E2%2B4%2Ax-12=%28x%2B6%29%28x-2%29.



Note: you can check the answer by expanding %28x%2B6%29%28x-2%29 to get x%5E2%2B4%2Ax-12 or by graphing the original expression and the answer (the two graphs should be identical).