how would i completely factor 8x^2-28x-60

Answer 1

The first step would be to factor a 4 from the expression so you have

8x² - 28x - 60 = 4(2x² -7x - 15)

Now, the problem becomes a matter of factoring 2x² - 7x - 15. Since the coefficient of the x² term is a 2, the only possibility is a factorization of the form

2x² - 7x - 15 = (2x + a)(x + b)

Where we have yet to determine a and b. To do so, we can perform the multiplication so we have

2x² - 7x - 15 = 2x² + 2bx + ax + ab = 2x² + (2b + a)x + ab

From this we can see that a and b must satisfy two conditions, namely

2b + a = -7

ab = -15

Solving for a in the first equation gives

a = -7 - 2b

Substituting that into the second equation yields

(-7 - 2b)(b) = -15

-2b² - 7b = -15

-2b² - 7b + 15 = 0

Using the quadratic formula, we get the integer solution b = -5. Using the relationship ab = -15, we see that a = 3. Therefore, we can put the a and b back into our solution such that

2x² - 7x - 15 = (2x + 3)(x - 5)

We cannot forget the 4 that we factored out at the very beginning.

Thus,

8x² - 28x - 60 = 4(2x + 3)(x - 5)