Factor 21x² - 100x + 100

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

a = 21
b = -100
c = 100

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

-10	-70x	100
3x	21x²	-30x
	7x	-10

We put 21x² (a) in the bottom left square and 100 (c) in the top right square, like this:

-10	-70x	100
3x	21x²	-30x
	7x	-10

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

More specifically, 21 times 100 is 2100. Therefore, we need to find the two numbers that multiply to equal 2100, and add to equal -100.

? × ? = 2100
? + ? = -100

After looking at this problem, we can see that the two numbers that multiply together to equal 2100, and add together to equal -100, are -70 and -30, as illustrated here:

-70 × -30 = 2100
-70 + -30 = -100

Now, we can fill in the last two squares in our box with -70x and -30x. Place -70x in the upper left square, and place -30x in the lower right square.

-10	-70x	100
3x	21x²	-30x
	7x	-10

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 -70x and 100. The greatest common factor of -70x and 100 is -10. Therefore, we write -10 to the left of the top row. You can see it here in the color green:

-10	-70x	100
3x	21x²	-30x
	7x	-10

Next, let’s look at the bottom row. We have the terms 21x² and -30x. The greatest common factor of 21x² and -30x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-10	-70x	100
3x	21x²	-30x
	7x	-10

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

21x² ÷ 3x = 7x

You can see this value colored in orange below:

-10	-70x	100
3x	21x²	-30x
	7x	-10

Next, we divide -30x by 3x (labeled in blue). This gives us -10.

-30x ÷ 3x = -10

You can see this value colored in purple below:

-10	-70x	100
3x	21x²	-30x
	7x	-10

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 + 100. 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:

(3x - 10)(7x - 10)

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

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