SOLUTION: Factor completely: 24x2 + 10x 4

Algebra ->  Expressions-with-variables -> SOLUTION: Factor completely: 24x2 + 10x 4       Log On

Ad: Mathway solves algebra homework problems with step-by-step help!


Question 111424: Factor completely:
24x2 + 10x 4

Answer by jim_thompson5910(33401) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

24%2Ax%5E2%2B10%2Ax-4 Start with the given expression.

2%2812x%5E2%2B5x-2%29 Factor out the GCF 2.

Now let's try to factor the inner expression 12x%5E2%2B5x-2


Looking at the expression 12x%5E2%2B5x-2, we can see that the first coefficient is 12, the second coefficient is 5, and the last term is -2.

Now multiply the first coefficient 12 by the last term -2 to get %2812%29%28-2%29=-24.

Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 5?

To find these two numbers, we need to list all of the factors of -24 (the previous product).

Factors of -24:



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

These factors pair up and multiply to -24.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:

First NumberSecond NumberSum

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

So the two numbers -3 and 8 both multiply to -24 and add to 5

Now replace the middle term 5x with -3x%2B8x. Remember, -3 and 8 add to 5. So this shows us that -3x%2B8x=5x.

12x%5E2%2Bhighlight%28-3x%2B8x%29-2 Replace the second term 5x with -3x%2B8x.

%2812x%5E2-3x%29%2B%288x-2%29 Group the terms into two pairs.

3x%284x-1%29%2B%288x-2%29 Factor out the GCF 3x from the first group.

3x%284x-1%29%2B2%284x-1%29 Factor out 2 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.

%283x%2B2%29%284x-1%29 Combine like terms. Or factor out the common term 4x-1


So 2%2812x%5E2%2B5x-2%29 then factors further to 2%283x%2B2%29%284x-1%29



So 24%2Ax%5E2%2B10%2Ax-4 completely factors to 2%283x%2B2%29%284x-1%29.

In other words, 24%2Ax%5E2%2B10%2Ax-4=2%283x%2B2%29%284x-1%29.

Note: you can check the answer by expanding 2%283x%2B2%29%284x-1%29 to get 24%2Ax%5E2%2B10%2Ax-4 or by graphing the original expression and the answer (the two graphs should be identical).