document.write( "Question 631560: how do i factor 8x^2-24x+18? i'm quite confused and can't seem to find a solution! \n" ); document.write( "

Algebra.Com's Answer #397665 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"8%2Ax%5E2-24%2Ax%2B18\"$ Start with the given expression.

$\"2%284x%5E2-12x%2B9%29\"$ Factor out the GCF $\"2\"$ .

Now let's try to factor the inner expression $\"4x%5E2-12x%2B9\"$

---------------------------------------------------------------

Looking at the expression $\"4x%5E2-12x%2B9\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"9\"$ .

Now multiply the first coefficient $\"4\"$ by the last term $\"9\"$ to get $\"%284%29%289%29=36\"$ .

Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?

To find these two numbers, we need to list all of the factors of $\"36\"$ (the previous product).

Factors of $\"36\"$ :

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36

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

These factors pair up and multiply to $\"36\"$ .

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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-12\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

From the table, we can see that the two numbers $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"-6\"$ both multiply to $\"36\"$ and add to $\"-12\"$

Now replace the middle term $\"-12x\"$ with $\"-6x-6x\"$ . Remember, $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ . So this shows us that $\"-6x-6x=-12x\"$ .

$\"4x%5E2%2Bhighlight%28-6x-6x%29%2B9\"$ Replace the second term $\"-12x\"$ with $\"-6x-6x\"$ .

$\"%284x%5E2-6x%29%2B%28-6x%2B9%29\"$ Group the terms into two pairs.

$\"2x%282x-3%29%2B%28-6x%2B9%29\"$ Factor out the GCF $\"2x\"$ from the first group.

$\"2x%282x-3%29-3%282x-3%29\"$ 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.

$\"%282x-3%29%282x-3%29\"$ Combine like terms. Or factor out the common term $\"2x-3\"$

$\"%282x-3%29%5E2\"$ Condense the terms.

--------------------------------------------------

So $\"2%284x%5E2-12x%2B9%29\"$ then factors further to $\"2%282x-3%29%5E2\"$

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

Answer:

So $\"8%2Ax%5E2-24%2Ax%2B18\"$ completely factors to $\"2%282x-3%29%5E2\"$ .

In other words, $\"8%2Ax%5E2-24%2Ax%2B18=2%282x-3%29%5E2\"$ .

Note: you can check the answer by expanding $\"2%282x-3%29%5E2\"$ to get $\"8%2Ax%5E2-24%2Ax%2B18\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( " \n" ); document.write( "

\n" );