SOLUTION: factoring a quadratic with leading coefficient greater than one y^2-10-39

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Question 91437: factoring a quadratic with leading coefficient greater than one
y^2-10-39
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression y%5E2-10y-39, we can see that the first coefficient is 1, the second coefficient is -10, and the last term is -39.



Now multiply the first coefficient 1 by the last term -39 to get %281%29%28-39%29=-39.



Now the question is: what two whole numbers multiply to -39 (the previous product) and add to the second coefficient -10?



To find these two numbers, we need to list all of the factors of -39 (the previous product).



Factors of -39:

1,3,13,39

-1,-3,-13,-39



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



These factors pair up and multiply to -39.

1*(-39) = -39
3*(-13) = -39
(-1)*(39) = -39
(-3)*(13) = -39


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -10:



First NumberSecond NumberSum
1-391+(-39)=-38
3-133+(-13)=-10
-139-1+39=38
-313-3+13=10




From the table, we can see that the two numbers 3 and -13 add to -10 (the middle coefficient).



So the two numbers 3 and -13 both multiply to -39 and add to -10



Now replace the middle term -10y with 3y-13y. Remember, 3 and -13 add to -10. So this shows us that 3y-13y=-10y.



y%5E2%2Bhighlight%283y-13y%29-39 Replace the second term -10y with 3y-13y.



%28y%5E2%2B3y%29%2B%28-13y-39%29 Group the terms into two pairs.



y%28y%2B3%29%2B%28-13y-39%29 Factor out the GCF y from the first group.



y%28y%2B3%29-13%28y%2B3%29 Factor out 13 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.



%28y-13%29%28y%2B3%29 Combine like terms. Or factor out the common term y%2B3



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Answer:



So y%5E2-10%2Ay-39 factors to %28y-13%29%28y%2B3%29.



In other words, y%5E2-10%2Ay-39=%28y-13%29%28y%2B3%29.



Note: you can check the answer by expanding %28y-13%29%28y%2B3%29 to get y%5E2-10%2Ay-39 or by graphing the original expression and the answer (the two graphs should be identical).