Factor -48x² - 90x

Factor -48x² - 90x - 27

Here we will show you how to factor the quadratic function -48x² - 90x - 27 using the box method. In other words, we will show you how to factor negative 48x squared minus 90x minus 27 (-48x^2 - 90x - 27) 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 48x² + 90x + 27. Now we can label the different parts of our equation, like this:

a = 48
b = 90
c = 27

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

9	18x	27
24x	48x²	72x
	2x	3

We put 48x² (a) in the bottom left square and 27 (c) in the top right square, like this:

9	18x	27
24x	48x²	72x
	2x	3

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 48 times 27 (a × c), and add together to equal 90 (b).

More specifically, 48 times 27 is 1296. Therefore, we need to find the two numbers that multiply to equal 1296, and add to equal 90.

? × ? = 1296
? + ? = 90

After looking at this problem, we can see that the two numbers that multiply together to equal 1296, and add together to equal 90, are 18 and 72, as illustrated here:

18 × 72 = 1296
18 + 72 = 90

Now, we can fill in the last two squares in our box with 18x and 72x. Place 18x in the upper left square, and place 72x in the lower right square.

9	18x	27
24x	48x²	72x
	2x	3

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 27. The greatest common factor of 18x and 27 is 9. Therefore, we write 9 to the left of the top row. You can see it here in the color green:

9	18x	27
24x	48x²	72x
	2x	3

Next, let’s look at the bottom row. We have the terms 48x² and 72x. The greatest common factor of 48x² and 72x is 24x. Therefore, we write 24x to the left of the bottom row. You can see it here in the color blue:

9	18x	27
24x	48x²	72x
	2x	3

To find the values below the table, we first divide 48x² by 24x (labeled in blue). This gives us 2x.

48x² ÷ 24x = 2x

You can see this value colored in orange below:

9	18x	27
24x	48x²	72x
	2x	3

Next, we divide 72x by 24x (labeled in blue). This gives us 3.

72x ÷ 24x = 3

You can see this value colored in purple below:

9	18x	27
24x	48x²	72x
	2x	3

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 -48x² - 90x - 27. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(24x + 9)(2x + 3)

In our original quadratic equation, -48x² - 90x - 27, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(24x + 9)(2x + 3)

That’s it! Now you know how to factor the equation -48x² - 90x - 27.

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

Factor -48x² - 90x + 12
Here is the next quadratic function on our list that we have factored for you.