Factoring a trinomial using decomposition

Worksheet:Quadratic Expand and Factor Skill Sheet

Review x^2+bx+c

Let’s expand (x+3)(x+4):


We notice that 3x+4x gives us the 7x; and that 3\times4 gives us the constant 12.

To factor a trinomial of the form x^2+bx+c, we need two values p,q such that p+q=b and p\times q=c.



The factor pairs of 54 are (1,54); (2,27); (3,18); (6,9). Now the last pair has a difference of 3, which is what we need for the middle term.

Let p=6 and q=-9 because 6+(-9)=-3 and 6 \times (-9)=-54.



Factor ax^2+bx+c, where a\ne1

Example 1:

First, let’s examine the expansion of a factored trinomial of this form:


This time, notice that 15+2=17 as expected, but 15 \times 2 \ne 10. Rather, 15\times 2 = 30.

Instead of looking for two numbers that add to 17 and multiply to 10, we need to look for two numbers that add to 17 and multiply to 30.

Example 2:



Look for two numbers, p and q, such that p+q=22 and p\times q = 5 \times 8=40.

The factor pairs of 40 are: (1,40); (2,20); (4,10); (5, 8)

To make 22, we need to use the pair (2,20).

We write a new line of work:


(It doesn’t matter if you write the 20x first or the 2x first).

Now we look for a common factor in the first two terms:


Perhaps suprisingly, (explained here), let’s look at the last two terms;


Ha! (x+4) is a common factor! Put together we have:


Taking out (x+4) as a common factor, we have:


Example 3:


Find two numbers, p and q such that p+q=-5 and p \times q = 4\times -6 = -24.

Factor pairs of 24 are (1,24); (2,12); (3,8); (4,6). Now 3-8=-5 so let p=3 and q=-8.

Decomposing the middle term we have:


Grouping we have:


Completing we have


Example 4:


First, we notice that all three terms share a common factor 3. Let’s factor this out:


Now we factor the reduced trinomial 2x^2+9x+4; however the 3 is kept present throughout. You might also try factoring without reducing to see how the result compares.

Now we look for p and q such that p+q=9, and p \times q = 2 \times 4 = 8.

Factor pairs of 8 are (1, 8); (2, 4). To also satisfy p+q=9, we need to use p=1 and q=8.

Decomposing we have


Grouping gives


Completing we have


Try it out:

Correct fields show as green, incorrect as red.

Geogebra link to the old applet

Geogebra link to this applet, for full screen capability

Proof of the method

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