SOLUTION: factor completely -15x^2-81x-30

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Question 704242: factor completely

Answer by jim_thompson5910(33401) About Me  (Show Source):
You can put this solution on YOUR website!

-15x%5E2-81x-30 Start with the given expression.

-3%285x%5E2%2B27x%2B10%29 Factor out the GCF -3.

Now let's try to factor the inner expression 5x%5E2%2B27x%2B10


Looking at the expression 5x%5E2%2B27x%2B10, we can see that the first coefficient is 5, the second coefficient is 27, and the last term is 10.

Now multiply the first coefficient 5 by the last term 10 to get %285%29%2810%29=50.

Now the question is: what two whole numbers multiply to 50 (the previous product) and add to the second coefficient 27?

To find these two numbers, we need to list all of the factors of 50 (the previous product).

Factors of 50:

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

These factors pair up and multiply to 50.
1*50 = 50
2*25 = 50
5*10 = 50
(-1)*(-50) = 50
(-2)*(-25) = 50
(-5)*(-10) = 50

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 27:

First NumberSecond NumberSum

From the table, we can see that the two numbers 2 and 25 add to 27 (the middle coefficient).

So the two numbers 2 and 25 both multiply to 50 and add to 27

Now replace the middle term 27x with 2x%2B25x. Remember, 2 and 25 add to 27. So this shows us that 2x%2B25x=27x.

5x%5E2%2Bhighlight%282x%2B25x%29%2B10 Replace the second term 27x with 2x%2B25x.

%285x%5E2%2B2x%29%2B%2825x%2B10%29 Group the terms into two pairs.

x%285x%2B2%29%2B%2825x%2B10%29 Factor out the GCF x from the first group.

x%285x%2B2%29%2B5%285x%2B2%29 Factor out 5 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%2B5%29%285x%2B2%29 Combine like terms. Or factor out the common term 5x%2B2


So -3%285x%5E2%2B27x%2B10%29 then factors further to -3%28x%2B5%29%285x%2B2%29



So -15x%5E2-81x-30 completely factors to -3%28x%2B5%29%285x%2B2%29.

In other words, -15x%5E2-81x-30=-3%28x%2B5%29%285x%2B2%29.

Note: you can check the answer by expanding -3%28x%2B5%29%285x%2B2%29 to get -15x%5E2-81x-30 or by graphing the original expression and the answer (the two graphs should be identical).