Factor 8x² - 2x

Factor 8x² - 2x - 21

Here we will show you how to factor the quadratic function 8x² - 2x - 21 using the box method. In other words, we will show you how to factor 8x squared minus 2x minus 21 (8x^2 - 2x - 21) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 8x² - 2x - 21, like this:

a = 8
b = -2
c = -21

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

-7	-14x	-21
4x	8x²	12x
	2x	3

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

-7	-14x	-21
4x	8x²	12x
	2x	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 -2 (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 -2.

? × ? = -168
? + ? = -2

After looking at this problem, we can see that the two numbers that multiply together to equal -168, and add together to equal -2, are -14 and 12, as illustrated here:

-14 × 12 = -168
-14 + 12 = -2

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

-7	-14x	-21
4x	8x²	12x
	2x	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 -14x and -21. The greatest common factor of -14x 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	-14x	-21
4x	8x²	12x
	2x	3

Next, let’s look at the bottom row. We have the terms 8x² and 12x. The greatest common factor of 8x² and 12x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-7	-14x	-21
4x	8x²	12x
	2x	3

To find the values below the table, we first divide 8x² by 4x (labeled in blue). This gives us 2x.

8x² ÷ 4x = 2x

You can see this value colored in orange below:

-7	-14x	-21
4x	8x²	12x
	2x	3

Next, we divide 12x by 4x (labeled in blue). This gives us 3.

12x ÷ 4x = 3

You can see this value colored in purple below:

-7	-14x	-21
4x	8x²	12x
	2x	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 - 2x - 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 the answer:

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

That’s it! Now you know how to factor the equation 8x² - 2x - 21.

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