Factor -2x² - 21x

Factor -2x² - 21x - 52

Here we will show you how to factor the quadratic function -2x² - 21x - 52 using the box method. In other words, we will show you how to factor negative 2x squared minus 21x minus 52 (-2x^2 - 21x - 52) 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 2x² + 21x + 52. Now we can label the different parts of our equation, like this:

a = 2
b = 21
c = 52

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

4	8x	52
x	2x²	13x
	2x	13

We put 2x² (a) in the bottom left square and 52 (c) in the top right square, like this:

4	8x	52
x	2x²	13x
	2x	13

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 2 times 52 (a × c), and add together to equal 21 (b).

More specifically, 2 times 52 is 104. Therefore, we need to find the two numbers that multiply to equal 104, and add to equal 21.

? × ? = 104
? + ? = 21

After looking at this problem, we can see that the two numbers that multiply together to equal 104, and add together to equal 21, are 8 and 13, as illustrated here:

8 × 13 = 104
8 + 13 = 21

Now, we can fill in the last two squares in our box with 8x and 13x. Place 8x in the upper left square, and place 13x in the lower right square.

4	8x	52
x	2x²	13x
	2x	13

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 8x and 52. The greatest common factor of 8x and 52 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	8x	52
x	2x²	13x
	2x	13

Next, let’s look at the bottom row. We have the terms 2x² and 13x. The greatest common factor of 2x² and 13x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

4	8x	52
x	2x²	13x
	2x	13

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

2x² ÷ x = 2x

You can see this value colored in orange below:

4	8x	52
x	2x²	13x
	2x	13

Next, we divide 13x by x (labeled in blue). This gives us 13.

13x ÷ x = 13

You can see this value colored in purple below:

4	8x	52
x	2x²	13x
	2x	13

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 -2x² - 21x - 52. 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 + 4)(2x + 13)

In our original quadratic equation, -2x² - 21x - 52, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 4)(2x + 13)

That’s it! Now you know how to factor the equation -2x² - 21x - 52.

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