Rewrite -5x^2-15x-7

Question

Rewrite

−5x^2−15x−7

in the form

(5x+5)×q(x)+r(x)

−5x^2−15x−7=(5x+5)×(__?__) + (__?___)

Answer 1

Dear Ginger,

To solve this problem, we would need to factor the equation -5x^2-15x-7 into the form (5x+5)*f(x)+g(x). I am using f and g, as the letters representing the polynomial, as they are the most used letter. The letter chosen is arbitrary though.

Let me clarify a few terms that will make it easier to understand my solution. I will call "-5x^2-15x-7" the solution, as we would need to use the form "(5x+5)*f(x)+g(x)" to expand into "-5x^2-15x-7".

First let us look at the form (5x+5)*f(x). We know that f(x) will be in the form ax+b. There is a resulting -5x^2, so we would need to multiply 5x by something (y) to get -5x^2. To find what value (y) should be, we divide -5x^2 by 5x. In doing so, we get -x. Therefore, we know the form will be (5x+5)*(-x+c)+g(x).

How do we know what c should be? Let us find that by expanding the form (5x+5)*(-x+c). Remember that to expand, you would need to do first, outer, inner, last.

First: 5x*(-x) = -5x^2

Outer: 5x*c = 5cx

Inner: 5*(-x) = -5x

Last: 5c

When expanded, we get -5x^2+5cx-5x+5c+g(x). By setting this equal to -5x^2-15x-7, we see that the -5x^2 match up. However, the -15x does not match:

-5x^2+5cx-5x+5c+g(x) = -5x^2-15x-7

Therefore, 5cx-5x = -15x. We set these equal because these are the parts with the x on both sides of the equation. Now we solve for what c should be.

5cx-5x = -15x

5cx = -15x + 5x

5cx = -10x

5c = -10

c = -2

Now that you know what c is, you can substitute c =-2 back into the equation. See below

-5x^2+5cx-5x+5c+g(x) = -5x^2-15x-7

-5x^2+5(-2)x-5x+5(-2)+g(x) = -5x^2-15x-7

-5x^2-10x-5x-10+g(x) = -5x^2-15x-7

-5x^2-15x-10+g(x) = -5x^2-15x-7

Notice that now we have made both sides of the equation have "-5x^2-15x". These are identical, so now we just need to balance the remaining part, -10+g(x) = -7

-10+g(x) = -7

g(x) = -7+10

g(x) = 3

Now we just need to combine everything we learned. We found that c=-2 and that g(x)=3. Earlier, we had the equation (5x+5)*(-x+c)+g(x). By substituting in c and g(x), we get (5x+5)*(-x-2)+3.

I hope my explanation helps! I would love to help you more in a class if you have any more questions!

-Eliam

Vanderbilt '26

Answer 2

−5x^2−15x−7=(5x+5)×(-x - 2) + (3)