Factor -11x² - 60x

Factor -11x² - 60x - 76

Here we will show you how to factor the quadratic function -11x² - 60x - 76 using the box method. In other words, we will show you how to factor negative 11x squared minus 60x minus 76 (-11x^2 - 60x - 76) 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 + 60x + 76. Now we can label the different parts of our equation, like this:

a = 11
b = 60
c = 76

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

2	22x	76
x	11x²	38x
	11x	38

We put 11x² (a) in the bottom left square and 76 (c) in the top right square, like this:

2	22x	76
x	11x²	38x
	11x	38

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 76 (a × c), and add together to equal 60 (b).

More specifically, 11 times 76 is 836. Therefore, we need to find the two numbers that multiply to equal 836, and add to equal 60.

? × ? = 836
? + ? = 60

After looking at this problem, we can see that the two numbers that multiply together to equal 836, and add together to equal 60, are 22 and 38, as illustrated here:

22 × 38 = 836
22 + 38 = 60

Now, we can fill in the last two squares in our box with 22x and 38x. Place 22x in the upper left square, and place 38x in the lower right square.

2	22x	76
x	11x²	38x
	11x	38

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 22x and 76. The greatest common factor of 22x and 76 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	22x	76
x	11x²	38x
	11x	38

Next, let’s look at the bottom row. We have the terms 11x² and 38x. The greatest common factor of 11x² and 38x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	22x	76
x	11x²	38x
	11x	38

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

11x² ÷ x = 11x

You can see this value colored in orange below:

2	22x	76
x	11x²	38x
	11x	38

Next, we divide 38x by x (labeled in blue). This gives us 38.

38x ÷ x = 38

You can see this value colored in purple below:

2	22x	76
x	11x²	38x
	11x	38

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 - 60x - 76. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(x + 2)(11x + 38)

In our original quadratic equation, -11x² - 60x - 76, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 2)(11x + 38)

That’s it! Now you know how to factor the equation -11x² - 60x - 76.

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