Factor 81x²

Factor 81x² - 100

Here we will show you how to factor the quadratic function 81x² - 100 using the box method. In other words, we will show you how to factor 81x squared minus 100 (81x^2 - 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 81x² - 100, like this:

a = 81
b = 0
c = -100

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

-10	-90x	-100
9x	81x²	90x
	9x	10

We put 81x² (a) in the bottom left square and -100 (c) in the top right square, like this:

-10	-90x	-100
9x	81x²	90x
	9x	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 81 times -100 (a × c), and add together to equal 0 (b).

More specifically, 81 times -100 is -8100. Therefore, we need to find the two numbers that multiply to equal -8100, and add to equal 0.

? × ? = -8100
? + ? = 0

After looking at this problem, we can see that the two numbers that multiply together to equal -8100, and add together to equal 0, are -90 and 90, as illustrated here:

-90 × 90 = -8100
-90 + 90 = 0

Now, we can fill in the last two squares in our box with -90x and 90x. Place -90x in the upper left square, and place 90x in the lower right square.

-10	-90x	-100
9x	81x²	90x
	9x	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 -90x and -100. The greatest common factor of -90x 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	-90x	-100
9x	81x²	90x
	9x	10

Next, let’s look at the bottom row. We have the terms 81x² and 90x. The greatest common factor of 81x² and 90x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

-10	-90x	-100
9x	81x²	90x
	9x	10

To find the values below the table, we first divide 81x² by 9x (labeled in blue). This gives us 9x.

81x² ÷ 9x = 9x

You can see this value colored in orange below:

-10	-90x	-100
9x	81x²	90x
	9x	10

Next, we divide 90x by 9x (labeled in blue). This gives us 10.

90x ÷ 9x = 10

You can see this value colored in purple below:

-10	-90x	-100
9x	81x²	90x
	9x	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 81x² - 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:

(9x - 10)(9x + 10)

That’s it! Now you know how to factor the equation 81x² - 100.

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