Factor Quadratics by Trial-and-Error

For the first time in this course, you’ll learn how to convert a quadratic that’s in its general form to a quadratic in factored form.

General form: y = ax² + bx + c   Factored form:  y = a(x – r)(x – s)   where r and s represent the x coordinates of the roots.

The first technique you can try is trial-and-error.

Take, for example, the quadratic equation:

x² – x – 6 = 0

First, identify the c constant and b coefficients. The c constant is -6, the b coefficient it -1. You have to do this all the time.

Next, you need to find two factors of -6 that multiply to it, and those same two factors add to -1.

The only possibility that words is -3 and +2.

-3 × (+2) = -6

-3 + 2 = -1

You then rewrite your equation as two factors: y = (x – 6)(x – 1).

Pointers to keep in mind:

  • Always begin the process of factoring a quadratic by common factoring if possible. In the example shown above, nothing could be common factored.
  • This technique can only be tried if the a coefficient is 1. Otherwise, another technique known as factoring by decomposition is used (next lesson).
  • Many quadratic expressions, such as x² + 3x + 5, cannot be factored over the integers. No two integers have a product of 5 and a sum of 3. In that case, we’ll use the quadratic formula (more to come on this later on).

Now let’s see this in action.