Factor -21x² - 60x

Factor -21x² - 60x - 36

Here we will show you how to factor the quadratic function -21x² - 60x - 36 using the box method. In other words, we will show you how to factor negative 21x squared minus 60x minus 36 (-21x^2 - 60x - 36) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 21x² + 60x + 36. Now we can label the different parts of our equation, like this:

a = 21
b = 60
c = 36

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

18	18x	36
21x	21x²	42x
	x	2

We put 21x² (a) in the bottom left square and 36 (c) in the top right square, like this:

18	18x	36
21x	21x²	42x
	x	2

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 21 times 36 (a × c), and add together to equal 60 (b).

More specifically, 21 times 36 is 756. Therefore, we need to find the two numbers that multiply to equal 756, and add to equal 60.

? × ? = 756
? + ? = 60

After looking at this problem, we can see that the two numbers that multiply together to equal 756, and add together to equal 60, are 18 and 42, as illustrated here:

18 × 42 = 756
18 + 42 = 60

Now, we can fill in the last two squares in our box with 18x and 42x. Place 18x in the upper left square, and place 42x in the lower right square.

18	18x	36
21x	21x²	42x
	x	2

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 36. The greatest common factor of 18x and 36 is 18. Therefore, we write 18 to the left of the top row. You can see it here in the color green:

18	18x	36
21x	21x²	42x
	x	2

Next, let’s look at the bottom row. We have the terms 21x² and 42x. The greatest common factor of 21x² and 42x is 21x. Therefore, we write 21x to the left of the bottom row. You can see it here in the color blue:

18	18x	36
21x	21x²	42x
	x	2

To find the values below the table, we first divide 21x² by 21x (labeled in blue). This gives us x.

21x² ÷ 21x = x

You can see this value colored in orange below:

18	18x	36
21x	21x²	42x
	x	2

Next, we divide 42x by 21x (labeled in blue). This gives us 2.

42x ÷ 21x = 2

You can see this value colored in purple below:

18	18x	36
21x	21x²	42x
	x	2

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 -21x² - 60x - 36. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(21x + 18)(x + 2)

In our original quadratic equation, -21x² - 60x - 36, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(21x + 18)(x + 2)

That’s it! Now you know how to factor the equation -21x² - 60x - 36.

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

Factor -21x² - 60x + 9
Here is the next quadratic function on our list that we have factored for you.