Factor 81x² - 81x + 20

Here we will show you how to factor the quadratic function 81x² - 81x + 20 using the box method. In other words, we will show you how to factor 81x squared minus 81x plus 20 (81x^2 - 81x + 20) 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 - 81x + 20, like this:

a = 81
b = -81
c = 20

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

-5	-45x	20
9x	81x²	-36x
	9x	-4

We put 81x² (a) in the bottom left square and 20 (c) in the top right square, like this:

-5	-45x	20
9x	81x²	-36x
	9x	-4

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 20 (a × c), and add together to equal -81 (b).

More specifically, 81 times 20 is 1620. Therefore, we need to find the two numbers that multiply to equal 1620, and add to equal -81.

? × ? = 1620
? + ? = -81

After looking at this problem, we can see that the two numbers that multiply together to equal 1620, and add together to equal -81, are -45 and -36, as illustrated here:

-45 × -36 = 1620
-45 + -36 = -81

Now, we can fill in the last two squares in our box with -45x and -36x. Place -45x in the upper left square, and place -36x in the lower right square.

-5	-45x	20
9x	81x²	-36x
	9x	-4

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 -45x and 20. The greatest common factor of -45x and 20 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-45x	20
9x	81x²	-36x
	9x	-4

Next, let’s look at the bottom row. We have the terms 81x² and -36x. The greatest common factor of 81x² and -36x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

-5	-45x	20
9x	81x²	-36x
	9x	-4

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:

-5	-45x	20
9x	81x²	-36x
	9x	-4

Next, we divide -36x by 9x (labeled in blue). This gives us -4.

-36x ÷ 9x = -4

You can see this value colored in purple below:

-5	-45x	20
9x	81x²	-36x
	9x	-4

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 - 81x + 20. 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 - 5)(9x - 4)

That’s it! Now you know how to factor the equation 81x² - 81x + 20.

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