SOLUTION: factor the algebraic expressions completely 12x^2 -34x + 24

Algebra ->  Expressions -> SOLUTION: factor the algebraic expressions completely 12x^2 -34x + 24      Log On


   

Question 476851: factor the algebraic expressions completely
12x^2 -34x + 24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

12x%5E2-34x%2B24 Start with the given expression.


2%286x%5E2-17x%2B12%29 Factor out the GCF 2.


Now let's try to factor the inner expression 6x%5E2-17x%2B12


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


Looking at the expression 6x%5E2-17x%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-9x. Remember, -8 and -9 add to -17. So this shows us that -8x-9x=-17x.


6x%5E2%2Bhighlight%28-8x-9x%29%2B12 Replace the second term -17x with -8x-9x.


%286x%5E2-8x%29%2B%28-9x%2B12%29 Group the terms into two pairs.


2x%283x-4%29%2B%28-9x%2B12%29 Factor out the GCF 2x from the first group.


2x%283x-4%29-3%283x-4%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%283x-4%29 Combine like terms. Or factor out the common term 3x-4


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


So 2%286x%5E2-17x%2B12%29 then factors further to 2%282x-3%29%283x-4%29


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


Answer:


So 12x%5E2-34x%2B24 completely factors to 2%282x-3%29%283x-4%29.


In other words, 12x%5E2-34x%2B24=2%282x-3%29%283x-4%29.


Note: you can check the answer by expanding 2%282x-3%29%283x-4%29 to get 12x%5E2-34x%2B24 or by graphing the original expression and the answer (the two graphs should be identical).