Factoring using the box method

Factoring using the box method is probably the best way to factor a trinomial of the form
ax2 + bx + c.

The box is just a 2x2 square that we use to put the terms of the trinomial.

Example #1

Factor 2x2 + 9x + 10 using the box method.

Step 1

First, put 2x2 and 10 in the box below as shown. You will always put the first term in the first row and the first column and the last term in the second row and second column.

Box method - first term and last term of the trinomial

Step 2

Multiply the first term by the last term: 2x2
×10 = 20x2

Look for factors of 20x2 that will add up to the second term or 9x.

20x2    =  1x 
× 20x
20x2  = 2x
× 10x
20x2  = 4x × 5x

Since 4x and 5x add up to 9x, put 4x and 5x in the box.

Box method - all terms

Step 3

Find the greatest common factor of each row and write them down on the left or right side of the box. Then, find the greatest common factor of each column and write them down on top or at the bottom of the box.

Box method

Notice the following:

2x times x = 2x2
2x times 2 = 4x
5 times x = 5x
5 times 2 = 10

The factors are (2x + 5)  and (x + 2), so 2x2 + 9x + 10 = (2x + 5) ×  (x + 2)

4x and 5x are interchangeable, so you could put 5x in the first row and 4x in the second row and still get the same answer.

Box method1


A common pitfall to avoid when factoring using the box method.

When factoring using the box method, you need to make sure that the greatest common factor of a, b, and c in ax2 + bx + c is equal to 1 as example #2 shows.

Example #2

Factor 6x2 + 27x + 30 using the box method.

Step 1

First put 6x2 and 30 in the box below as shown.

Step 2

Multiply the first term by the last term: 6x2
× 30 = 180x2

Look for factors of 180x2 that will add up to 27x.

Since 12x
× 15x = 180x2 and 12x + 15x = 27x, put 12x and 15x in the box.

Step 3

Find the greatest common factor of each row and write them down on the left or right side of the box. Then, find the greatest common factor of each column and write them down on top or at the bottom of the box. We end up with the following box.

Factoring using the box method

However, we have a little problem . 6x times 3x is not equal to 6x2, 6x times 6 is not equal to 12x, 15 times 3x is not equal to 15x, and 15 times 6 is not equal to 30.

In order for us to see what happened here, we need to make an important observation.

Notice that 6x2 + 27x + 30 = 3(2x2 + 9x + 10 ) and we already factored 2x2 + 9x + 10

2x2 + 9x + 10 =  (2x + 5)× (x + 2)

Therefore, 6x2 + 27x + 30 = 3(2x2 + 9x + 10 ) = 3(2x + 5)×(x + 2)

Looking at the box immediately above again, we see that we can factor the entire thing by 3. And if we do that we will get the same answer as illustrated below.

Factoring 6x^2 + 27x + 30


Look at the image immediately above again. The box inside the parentheses is the box that we used to factor 2x2 + 9x + 10.

Therefore when factoring using the box method, make sure you factor the trinomial ax2 + bx + c until the greatest common factor of a, b, and c is equal to 1 to avoid complicating things.

Example #3

Factor 4x2 - 8x - 12 using the box method.

First, factor 4x2 - 8x - 12 using the greatest common factor.

4x2 - 8x - 12 = 4(x2 - 2x - 3)

Now, you just need to factor x2 - 2x - 3 using the box method

Step 1

First put x2 and -3 in the box

Step 2

Multiply the first term by the last term: x2
× -3 = -3x2

Look for factors of -3x2 that will add up to -2x.

Since -3x
× x = -3x2 and -3x + x = -2x put -3x and x in the box.

Step 3

Find the greatest common factor of each row and write them down on the left or right side of the box. Then, find the greatest common factor of each column and write them down on top or at the bottom of the box. We end up with the following box.

Factoring x^2 - 2x - 3 using the box method

x2 - 2x - 3 = (x + 1)(x - 3)

4x2 - 8x - 12 = 4(x2 - 2x - 3) = 4(x + 1)(x - 3)

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