Factoring: Box & Diamond Method

The box and diamond method makes use of the shapes below to find the factored form of a quadratic function using the standard form.

Remember that the standard form of our quadratic function is:

The box and diamond method is a step by step process to find the factored form. The first step of our method is to make sure our quadratic function is in the same order and has 3 terms like the standard form above. If one of the terms is missing, you can replace it with a zero. For example:

Then we fill in the boxes as follows.

Let's try it with a real equation, we will use the equation:

Now to find the sides of the diamond. To do this, first we have to find all the factors of 6. The factors of 6 are 1 and 6, and 2 and 3. Now we have to see which of these add to 5. We can see that 2 and 3 add to 5. Therefore, we will put 2x and 3x into the sides of the diamond. We also can put in 2x and 3x into the remaining boxes in the box portion.

Next, we find the common multiples for the box. We can see that both of the boxes in the top row are divisible by 2 and both the boxes in the second row are divisible by x. So, we fill in the the box like so:

The Second to last step is to find the common divisors for each column. We can see in the first column, both boxes are divisible by x. We can see in the second column that both boxes are divisible by 3. So we fill in the box like so:

Finally, our factors are the parts outside of the box, so we write the factors out as:

(x + 2)(x + 3)

Remember that the factored form of the equation is equivalent to the standard form of the equation.