Factor 21x² - 100x + 64

Here we will show you how to factor the quadratic function 21x² - 100x + 64 using the box method. In other words, we will show you how to factor 21x squared minus 100x plus 64 (21x^2 - 100x + 64) 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 21x² - 100x + 64, like this:

a = 21
b = -100
c = 64

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

-4	-84x	64
x	21x²	-16x
	21x	-16

We put 21x² (a) in the bottom left square and 64 (c) in the top right square, like this:

-4	-84x	64
x	21x²	-16x
	21x	-16

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 64 (a × c), and add together to equal -100 (b).

More specifically, 21 times 64 is 1344. Therefore, we need to find the two numbers that multiply to equal 1344, and add to equal -100.

? × ? = 1344
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 1344, and add together to equal -100, are -84 and -16, as illustrated here:

-84 × -16 = 1344
-84 + -16 = -100

Now, we can fill in the last two squares in our box with -84x and -16x. Place -84x in the upper left square, and place -16x in the lower right square.

-4	-84x	64
x	21x²	-16x
	21x	-16

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 -84x and 64. The greatest common factor of -84x and 64 is -4. Therefore, we write -4 to the left of the top row. You can see it here in the color green:

-4	-84x	64
x	21x²	-16x
	21x	-16

Next, let’s look at the bottom row. We have the terms 21x² and -16x. The greatest common factor of 21x² and -16x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-4	-84x	64
x	21x²	-16x
	21x	-16

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

21x² ÷ x = 21x

You can see this value colored in orange below:

-4	-84x	64
x	21x²	-16x
	21x	-16

Next, we divide -16x by x (labeled in blue). This gives us -16.

-16x ÷ x = -16

You can see this value colored in purple below:

-4	-84x	64
x	21x²	-16x
	21x	-16

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 - 100x + 64. 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:

(x - 4)(21x - 16)

That’s it! Now you know how to factor the equation 21x² - 100x + 64.

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