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 × cand whose sum isb. 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 **a**x² + **b**x + **c** can be factored over the integers. The trinomial factorability test is shown below: