Factor 24x² - 71x + 21

Here we will show you how to factor the quadratic function 24x² - 71x + 21 using the box method. In other words, we will show you how to factor 24x squared minus 71x plus 21 (24x^2 - 71x + 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 24x² - 71x + 21, like this:

a = 24
b = -71
c = 21

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

-21	-63x	21
8x	24x²	-8x
	3x	-1

We put 24x² (a) in the bottom left square and 21 (c) in the top right square, like this:

-21	-63x	21
8x	24x²	-8x
	3x	-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 24 times 21 (a × c), and add together to equal -71 (b).

More specifically, 24 times 21 is 504. Therefore, we need to find the two numbers that multiply to equal 504, and add to equal -71.

? × ? = 504
? + ? = -71

After looking at this problem, we can see that the two numbers that multiply together to equal 504, and add together to equal -71, are -63 and -8, as illustrated here:

-63 × -8 = 504
-63 + -8 = -71

Now, we can fill in the last two squares in our box with -63x and -8x. Place -63x in the upper left square, and place -8x in the lower right square.

-21	-63x	21
8x	24x²	-8x
	3x	-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 -63x and 21. The greatest common factor of -63x and 21 is -21. Therefore, we write -21 to the left of the top row. You can see it here in the color green:

-21	-63x	21
8x	24x²	-8x
	3x	-1

Next, let’s look at the bottom row. We have the terms 24x² and -8x. The greatest common factor of 24x² and -8x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

-21	-63x	21
8x	24x²	-8x
	3x	-1

To find the values below the table, we first divide 24x² by 8x (labeled in blue). This gives us 3x.

24x² ÷ 8x = 3x

You can see this value colored in orange below:

-21	-63x	21
8x	24x²	-8x
	3x	-1

Next, we divide -8x by 8x (labeled in blue). This gives us -1.

-8x ÷ 8x = -1

You can see this value colored in purple below:

-21	-63x	21
8x	24x²	-8x
	3x	-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 24x² - 71x + 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:

(8x - 21)(3x - 1)

That’s it! Now you know how to factor the equation 24x² - 71x + 21.

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