Factor -11x² - 12x

Factor -11x² - 12x - 1

Here we will show you how to factor the quadratic function -11x² - 12x - 1 using the box method. In other words, we will show you how to factor negative 11x squared minus 12x minus 1 (-11x^2 - 12x - 1) 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 11x² + 12x + 1. Now we can label the different parts of our equation, like this:

a = 11
b = 12
c = 1

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

1	x	1
11x	11x²	11x
	x	1

We put 11x² (a) in the bottom left square and 1 (c) in the top right square, like this:

1	x	1
11x	11x²	11x
	x	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 11 times 1 (a × c), and add together to equal 12 (b).

More specifically, 11 times 1 is 11. Therefore, we need to find the two numbers that multiply to equal 11, and add to equal 12.

? × ? = 11
? + ? = 12

After looking at this problem, we can see that the two numbers that multiply together to equal 11, and add together to equal 12, are 1 and 11, as illustrated here:

1 × 11 = 11
1 + 11 = 12

Now, we can fill in the last two squares in our box with x and 11x. Place x in the upper left square, and place 11x in the lower right square.

1	x	1
11x	11x²	11x
	x	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 x and 1. The greatest common factor of x and 1 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	x	1
11x	11x²	11x
	x	1

Next, let’s look at the bottom row. We have the terms 11x² and 11x. The greatest common factor of 11x² and 11x is 11x. Therefore, we write 11x to the left of the bottom row. You can see it here in the color blue:

1	x	1
11x	11x²	11x
	x	1

To find the values below the table, we first divide 11x² by 11x (labeled in blue). This gives us x.

11x² ÷ 11x = x

You can see this value colored in orange below:

1	x	1
11x	11x²	11x
	x	1

Next, we divide 11x by 11x (labeled in blue). This gives us 1.

11x ÷ 11x = 1

You can see this value colored in purple below:

1	x	1
11x	11x²	11x
	x	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 -11x² - 12x - 1. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(11x + 1)(x + 1)

In our original quadratic equation, -11x² - 12x - 1, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(11x + 1)(x + 1)

That’s it! Now you know how to factor the equation -11x² - 12x - 1.

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