Factor 36x² - 100x + 56

Here we will show you how to factor the quadratic function 36x² - 100x + 56 using the box method. In other words, we will show you how to factor 36x squared minus 100x plus 56 (36x^2 - 100x + 56) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 36x² - 100x + 56, like this:

a = 36
b = -100
c = 56

Step 2: Next, we need to draw a box and divide it into four squares:

-8	-72x	56
4x	36x²	-28x
	9x	-7

We put 36x² (a) in the bottom left square and 56 (c) in the top right square, like this:

-8	-72x	56
4x	36x²	-28x
	9x	-7

Step 3: In order to fill in the other two squares, we need to do some math. We have to find two numbers that multiply together to give us the product of 36 times 56 (a × c), and add together to equal -100 (b).

More specifically, 36 times 56 is 2016. Therefore, we need to find the two numbers that multiply to equal 2016, and add to equal -100.

? × ? = 2016
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 2016, and add together to equal -100, are -72 and -28, as illustrated here:

-72 × -28 = 2016
-72 + -28 = -100

Now, we can fill in the last two squares in our box with -72x and -28x. Place -72x in the upper left square, and place -28x in the lower right square.

-8	-72x	56
4x	36x²	-28x
	9x	-7

Step 4: Next, we need to find four final numbers to finish factoring our equation. We can see that our box is a 2 x 2 table made up of rows and columns.

Our goal is to find the numbers that, when multiplied together, result in the products in each square. We can do this by finding the greatest common factor in each row.

Let’s look at the top row. We have the terms -72x and 56. The greatest common factor of -72x and 56 is -8. Therefore, we write -8 to the left of the top row. You can see it here in the color green:

-8	-72x	56
4x	36x²	-28x
	9x	-7

Next, let’s look at the bottom row. We have the terms 36x² and -28x. The greatest common factor of 36x² and -28x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-8	-72x	56
4x	36x²	-28x
	9x	-7

To find the values below the table, we first divide 36x² by 4x (labeled in blue). This gives us 9x.

36x² ÷ 4x = 9x

You can see this value colored in orange below:

-8	-72x	56
4x	36x²	-28x
	9x	-7

Next, we divide -28x by 4x (labeled in blue). This gives us -7.

-28x ÷ 4x = -7

You can see this value colored in purple below:

-8	-72x	56
4x	36x²	-28x
	9x	-7

Step 5: Now we have all of the information we need to finish factoring the equation. The values outside of the box are used to factor 36x² - 100x + 56. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(4x - 8)(9x - 7)

That’s it! Now you know how to factor the equation 36x² - 100x + 56.

Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor 36x² - 100x + 64
Here is the next quadratic function on our list that we have factored for you.