SOLUTION: How would I factor "3k^2-24k-60" completely?

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Question 894960: How would I factor "3k^2-24k-60" completely?
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)


3%2Ak%5E2-24%2Ak-60 Start with the given expression.



3%28k%5E2-8k-20%29 Factor out the GCF 3.



Now let's try to factor the inner expression k%5E2-8k-20



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Looking at the expression k%5E2-8k-20, we can see that the first coefficient is 1, the second coefficient is -8, and the last term is -20.



Now multiply the first coefficient 1 by the last term -20 to get %281%29%28-20%29=-20.



Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient -8?



To find these two numbers, we need to list all of the factors of -20 (the previous product).



Factors of -20:

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20



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



These factors pair up and multiply to -20.

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -8:



First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1




From the table, we can see that the two numbers 2 and -10 add to -8 (the middle coefficient).



So the two numbers 2 and -10 both multiply to -20 and add to -8



Now replace the middle term -8k with 2k-10k. Remember, 2 and -10 add to -8. So this shows us that 2k-10k=-8k.



k%5E2%2Bhighlight%282k-10k%29-20 Replace the second term -8k with 2k-10k.



%28k%5E2%2B2k%29%2B%28-10k-20%29 Group the terms into two pairs.



k%28k%2B2%29%2B%28-10k-20%29 Factor out the GCF k from the first group.



k%28k%2B2%29-10%28k%2B2%29 Factor out 10 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.



%28k-10%29%28k%2B2%29 Combine like terms. Or factor out the common term k%2B2



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So 3%28k%5E2-8k-20%29 then factors further to 3%28k-10%29%28k%2B2%29



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Answer:



So 3%2Ak%5E2-24%2Ak-60 completely factors to 3%28k-10%29%28k%2B2%29.



In other words, 3%2Ak%5E2-24%2Ak-60=3%28k-10%29%28k%2B2%29.



Note: you can check the answer by expanding 3%28k-10%29%28k%2B2%29 to get 3%2Ak%5E2-24%2Ak-60 or by graphing the original expression and the answer (the two graphs should be identical).