SOLUTION: Factor completely. Show all work necessary. 6x^2 + 17x +12

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Question 281983: Factor completely. Show all work necessary.
6x^2 + 17x +12
Answer by richwmiller(17219) 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 6x%5E2%2B17x%2B12, we can see that the first coefficient is 6, the second coefficient is 17, and the last term is 12.



Now multiply the first coefficient 6 by the last term 12 to get %286%29%2812%29=72.



Now the question is: what two whole numbers multiply to 72 (the previous product) and add to the second coefficient 17?



To find these two numbers, we need to list all of the factors of 72 (the previous product).



Factors of 72:

1,2,3,4,6,8,9,12,18,24,36,72

-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72



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



These factors pair up and multiply to 72.

1*72 = 72
2*36 = 72
3*24 = 72
4*18 = 72
6*12 = 72
8*9 = 72
(-1)*(-72) = 72
(-2)*(-36) = 72
(-3)*(-24) = 72
(-4)*(-18) = 72
(-6)*(-12) = 72
(-8)*(-9) = 72


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 17:



First NumberSecond NumberSum
1721+72=73
2362+36=38
3243+24=27
4184+18=22
6126+12=18
898+9=17
-1-72-1+(-72)=-73
-2-36-2+(-36)=-38
-3-24-3+(-24)=-27
-4-18-4+(-18)=-22
-6-12-6+(-12)=-18
-8-9-8+(-9)=-17




From the table, we can see that the two numbers 8 and 9 add to 17 (the middle coefficient).



So the two numbers 8 and 9 both multiply to 72 and add to 17



Now replace the middle term 17x with 8x%2B9x. Remember, 8 and 9 add to 17. So this shows us that 8x%2B9x=17x.



6x%5E2%2Bhighlight%288x%2B9x%29%2B12 Replace the second term 17x with 8x%2B9x.



%286x%5E2%2B8x%29%2B%289x%2B12%29 Group the terms into two pairs.



2x%283x%2B4%29%2B%289x%2B12%29 Factor out the GCF 2x from the first group.



2x%283x%2B4%29%2B3%283x%2B4%29 Factor out 3 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.



%282x%2B3%29%283x%2B4%29 Combine like terms. Or factor out the common term 3x%2B4



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



Answer:



So 6%2Ax%5E2%2B17%2Ax%2B12 factors to %282x%2B3%29%283x%2B4%29.



In other words, 6%2Ax%5E2%2B17%2Ax%2B12=%282x%2B3%29%283x%2B4%29.



Note: you can check the answer by expanding %282x%2B3%29%283x%2B4%29 to get 6%2Ax%5E2%2B17%2Ax%2B12 or by graphing the original expression and the answer (the two graphs should be identical).


Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B17x%2B12+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2817%29%5E2-4%2A6%2A12=1.

Discriminant d=1 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-17%2B-sqrt%28+1+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2817%29%2Bsqrt%28+1+%29%29%2F2%5C6+=+-1.33333333333333
x%5B2%5D+=+%28-%2817%29-sqrt%28+1+%29%29%2F2%5C6+=+-1.5

Quadratic expression 6x%5E2%2B17x%2B12 can be factored:
6x%5E2%2B17x%2B12+=+6%28x--1.33333333333333%29%2A%28x--1.5%29
Again, the answer is: -1.33333333333333, -1.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B17%2Ax%2B12+%29