Factor -16x² - 56x

Factor -16x² - 56x - 45

Here we will show you how to factor the quadratic function -16x² - 56x - 45 using the box method. In other words, we will show you how to factor negative 16x squared minus 56x minus 45 (-16x^2 - 56x - 45) 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 16x² + 56x + 45. Now we can label the different parts of our equation, like this:

a = 16
b = 56
c = 45

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

5	20x	45
4x	16x²	36x
	4x	9

We put 16x² (a) in the bottom left square and 45 (c) in the top right square, like this:

5	20x	45
4x	16x²	36x
	4x	9

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 16 times 45 (a × c), and add together to equal 56 (b).

More specifically, 16 times 45 is 720. Therefore, we need to find the two numbers that multiply to equal 720, and add to equal 56.

? × ? = 720
? + ? = 56

After looking at this problem, we can see that the two numbers that multiply together to equal 720, and add together to equal 56, are 20 and 36, as illustrated here:

20 × 36 = 720
20 + 36 = 56

Now, we can fill in the last two squares in our box with 20x and 36x. Place 20x in the upper left square, and place 36x in the lower right square.

5	20x	45
4x	16x²	36x
	4x	9

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 20x and 45. The greatest common factor of 20x and 45 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	20x	45
4x	16x²	36x
	4x	9

Next, let’s look at the bottom row. We have the terms 16x² and 36x. The greatest common factor of 16x² and 36x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

5	20x	45
4x	16x²	36x
	4x	9

To find the values below the table, we first divide 16x² by 4x (labeled in blue). This gives us 4x.

16x² ÷ 4x = 4x

You can see this value colored in orange below:

5	20x	45
4x	16x²	36x
	4x	9

Next, we divide 36x by 4x (labeled in blue). This gives us 9.

36x ÷ 4x = 9

You can see this value colored in purple below:

5	20x	45
4x	16x²	36x
	4x	9

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 -16x² - 56x - 45. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x + 5)(4x + 9)

In our original quadratic equation, -16x² - 56x - 45, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 5)(4x + 9)

That’s it! Now you know how to factor the equation -16x² - 56x - 45.

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

Factor -16x² - 56x - 40
Here is the next quadratic function on our list that we have factored for you.