Factor -8x² - 31x

Factor -8x² - 31x - 21

Here we will show you how to factor the quadratic function -8x² - 31x - 21 using the box method. In other words, we will show you how to factor negative 8x squared minus 31x minus 21 (-8x^2 - 31x - 21) 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 8x² + 31x + 21. Now we can label the different parts of our equation, like this:

a = 8
b = 31
c = 21

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

7	7x	21
8x	8x²	24x
	x	3

We put 8x² (a) in the bottom left square and 21 (c) in the top right square, like this:

7	7x	21
8x	8x²	24x
	x	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 8 times 21 (a × c), and add together to equal 31 (b).

More specifically, 8 times 21 is 168. Therefore, we need to find the two numbers that multiply to equal 168, and add to equal 31.

? × ? = 168
? + ? = 31

After looking at this problem, we can see that the two numbers that multiply together to equal 168, and add together to equal 31, are 7 and 24, as illustrated here:

7 × 24 = 168
7 + 24 = 31

Now, we can fill in the last two squares in our box with 7x and 24x. Place 7x in the upper left square, and place 24x in the lower right square.

7	7x	21
8x	8x²	24x
	x	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 7x and 21. The greatest common factor of 7x and 21 is 7. Therefore, we write 7 to the left of the top row. You can see it here in the color green:

7	7x	21
8x	8x²	24x
	x	3

Next, let’s look at the bottom row. We have the terms 8x² and 24x. The greatest common factor of 8x² and 24x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

7	7x	21
8x	8x²	24x
	x	3

To find the values below the table, we first divide 8x² by 8x (labeled in blue). This gives us x.

8x² ÷ 8x = x

You can see this value colored in orange below:

7	7x	21
8x	8x²	24x
	x	3

Next, we divide 24x by 8x (labeled in blue). This gives us 3.

24x ÷ 8x = 3

You can see this value colored in purple below:

7	7x	21
8x	8x²	24x
	x	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 -8x² - 31x - 21. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(8x + 7)(x + 3)

In our original quadratic equation, -8x² - 31x - 21, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(8x + 7)(x + 3)

That’s it! Now you know how to factor the equation -8x² - 31x - 21.

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