Factor -21x² - 26x

Factor -21x² - 26x - 8

Here we will show you how to factor the quadratic function -21x² - 26x - 8 using the box method. In other words, we will show you how to factor negative 21x squared minus 26x minus 8 (-21x^2 - 26x - 8) 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 21x² + 26x + 8. Now we can label the different parts of our equation, like this:

a = 21
b = 26
c = 8

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

4	12x	8
7x	21x²	14x
	3x	2

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

4	12x	8
7x	21x²	14x
	3x	2

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 21 times 8 (a × c), and add together to equal 26 (b).

More specifically, 21 times 8 is 168. Therefore, we need to find the two numbers that multiply to equal 168, and add to equal 26.

? × ? = 168
? + ? = 26

After looking at this problem, we can see that the two numbers that multiply together to equal 168, and add together to equal 26, are 12 and 14, as illustrated here:

12 × 14 = 168
12 + 14 = 26

Now, we can fill in the last two squares in our box with 12x and 14x. Place 12x in the upper left square, and place 14x in the lower right square.

4	12x	8
7x	21x²	14x
	3x	2

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 12x and 8. The greatest common factor of 12x and 8 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	12x	8
7x	21x²	14x
	3x	2

Next, let’s look at the bottom row. We have the terms 21x² and 14x. The greatest common factor of 21x² and 14x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

4	12x	8
7x	21x²	14x
	3x	2

To find the values below the table, we first divide 21x² by 7x (labeled in blue). This gives us 3x.

21x² ÷ 7x = 3x

You can see this value colored in orange below:

4	12x	8
7x	21x²	14x
	3x	2

Next, we divide 14x by 7x (labeled in blue). This gives us 2.

14x ÷ 7x = 2

You can see this value colored in purple below:

4	12x	8
7x	21x²	14x
	3x	2

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 -21x² - 26x - 8. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(7x + 4)(3x + 2)

In our original quadratic equation, -21x² - 26x - 8, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(7x + 4)(3x + 2)

That’s it! Now you know how to factor the equation -21x² - 26x - 8.

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