Can anyone explain how to solve quadratic equations by factoring?

Question

I am having troubles understanding how to solve quadratic equations by factoring.

Answer 1

You want to find two numbers that multiply to give you the constant term in your quadratic and combine to give you your x coefficient. However, this only works for quadratics of the form: x^2+ax+b where the coefficient of x^2 is 1. So, for example, if you want to factor x^2+7x+12=0 then because 4 and 3 multiply to give you 12 and combine to give you 7 you get: (x+4)(x+3)=0 as a factorization.

Answer 2

Factoring is really trial and error.
Consider the equation 6x^2 + 3x - 18 = 0.
First, notice that you can factor out 3, to reduce the numbers:
3(2x^2 + x - 6) = 0.
The remaining factors will be of the form (ax + b)(cx - d) = 0.
The coefficient of x^2 is ac, and the absolute term is -bd.
The coefficient of x is bc - ad.
You now have to try pairs of possible factors for 2 and for 6 until you find the one such that bc - ad = 1.
When you have found those, you have the factors:
3(2x - 3)(x + 2).