Factor 36x²

Factor 36x² - 9

Here we will show you how to factor the quadratic function 36x² - 9 using the box method. In other words, we will show you how to factor 36x squared minus 9 (36x^2 - 9) 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 - 9, like this:

a = 36
b = 0
c = -9

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

-9	-18x	-9
18x	36x²	18x
	2x	1

We put 36x² (a) in the bottom left square and -9 (c) in the top right square, like this:

-9	-18x	-9
18x	36x²	18x
	2x	1

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 -9 (a × c), and add together to equal 0 (b).

More specifically, 36 times -9 is -324. Therefore, we need to find the two numbers that multiply to equal -324, and add to equal 0.

? × ? = -324
? + ? = 0

After looking at this problem, we can see that the two numbers that multiply together to equal -324, and add together to equal 0, are -18 and 18, as illustrated here:

-18 × 18 = -324
-18 + 18 = 0

Now, we can fill in the last two squares in our box with -18x and 18x. Place -18x in the upper left square, and place 18x in the lower right square.

-9	-18x	-9
18x	36x²	18x
	2x	1

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 -18x and -9. The greatest common factor of -18x and -9 is -9. Therefore, we write -9 to the left of the top row. You can see it here in the color green:

-9	-18x	-9
18x	36x²	18x
	2x	1

Next, let’s look at the bottom row. We have the terms 36x² and 18x. The greatest common factor of 36x² and 18x is 18x. Therefore, we write 18x to the left of the bottom row. You can see it here in the color blue:

-9	-18x	-9
18x	36x²	18x
	2x	1

To find the values below the table, we first divide 36x² by 18x (labeled in blue). This gives us 2x.

36x² ÷ 18x = 2x

You can see this value colored in orange below:

-9	-18x	-9
18x	36x²	18x
	2x	1

Next, we divide 18x by 18x (labeled in blue). This gives us 1.

18x ÷ 18x = 1

You can see this value colored in purple below:

-9	-18x	-9
18x	36x²	18x
	2x	1

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 - 9. 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:

(18x - 9)(2x + 1)

That’s it! Now you know how to factor the equation 36x² - 9.

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