SOLUTION: 24x2-42x-12=0

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Question 590972: 24x2-42x-12=0
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)


24%2Ax%5E2-42%2Ax-12 Start with the given expression.



6%284x%5E2-7x-2%29 Factor out the GCF 6.



Now let's try to factor the inner expression 4x%5E2-7x-2



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Looking at the expression 4x%5E2-7x-2, we can see that the first coefficient is 4, the second coefficient is -7, and the last term is -2.



Now multiply the first coefficient 4 by the last term -2 to get %284%29%28-2%29=-8.



Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient -7?



To find these two numbers, we need to list all of the factors of -8 (the previous product).



Factors of -8:

1,2,4,8

-1,-2,-4,-8



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



These factors pair up and multiply to -8.

1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -7:



First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2




From the table, we can see that the two numbers 1 and -8 add to -7 (the middle coefficient).



So the two numbers 1 and -8 both multiply to -8 and add to -7



Now replace the middle term -7x with x-8x. Remember, 1 and -8 add to -7. So this shows us that x-8x=-7x.



4x%5E2%2Bhighlight%28x-8x%29-2 Replace the second term -7x with x-8x.



%284x%5E2%2Bx%29%2B%28-8x-2%29 Group the terms into two pairs.



x%284x%2B1%29%2B%28-8x-2%29 Factor out the GCF x from the first group.



x%284x%2B1%29-2%284x%2B1%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.



%28x-2%29%284x%2B1%29 Combine like terms. Or factor out the common term 4x%2B1



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So 6%284x%5E2-7x-2%29 then factors further to 6%28x-2%29%284x%2B1%29



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Answer:



So 24%2Ax%5E2-42%2Ax-12 completely factors to 6%28x-2%29%284x%2B1%29.



In other words, 24%2Ax%5E2-42%2Ax-12=6%28x-2%29%284x%2B1%29.



Note: you can check the answer by expanding 6%28x-2%29%284x%2B1%29 to get 24%2Ax%5E2-42%2Ax-12 or by graphing the original expression and the answer (the two graphs should be identical).