Factor Quadratics by Decomposition

When a general form quadratic has an a coefficient greater than 1, the trial-and-error method no longer works. Take, for example, the equation:

y = 3x² + 5x + 6

You can’t choose 3 and 2 as factors that multiply to 6 and add to 5 – it doesn’t work that way.

Arguably you could common factor the 3, leaving x² with a coefficient of 1:

y = 3 ( x² + 5/3x + 2 )

But then you’re left with finding two factors of 2 that add to 5/3!

To factor quadratics whose a > 1,  we use a technique known as factoring by decomposition, which involving breaking up the middle term – hence the name.

Let’s see a few examples of this technique in action.

To summarize, factoring by decomposition involves finding two integers whose product is a × c and whose sum is b. Then, break up the middle term and factor by grouping.

Interestingly, referring back to the initial equation:

y = 3x² + 5x + 6

If you try factoring by decomposition here, it still won’t yield a factored-form quadratic. In that case, you’d have to use the quadratic formula to find the roots (more on this to come). Therefore, not all quadratic expressions of the form ax² + bx + c can be factored over the integers. The trinomial factorability test is shown below: