Factor 90x² - 97x + 21

Here we will show you how to factor the quadratic function 90x² - 97x + 21 using the box method. In other words, we will show you how to factor 90x squared minus 97x plus 21 (90x^2 - 97x + 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 90x² - 97x + 21, like this:

a = 90
b = -97
c = 21

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

-7	-70x	21
9x	90x²	-27x
	10x	-3

We put 90x² (a) in the bottom left square and 21 (c) in the top right square, like this:

-7	-70x	21
9x	90x²	-27x
	10x	-3

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 90 times 21 (a × c), and add together to equal -97 (b).

More specifically, 90 times 21 is 1890. Therefore, we need to find the two numbers that multiply to equal 1890, and add to equal -97.

? × ? = 1890
? + ? = -97

After looking at this problem, we can see that the two numbers that multiply together to equal 1890, and add together to equal -97, are -70 and -27, as illustrated here:

-70 × -27 = 1890
-70 + -27 = -97

Now, we can fill in the last two squares in our box with -70x and -27x. Place -70x in the upper left square, and place -27x in the lower right square.

-7	-70x	21
9x	90x²	-27x
	10x	-3

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 21. The greatest common factor of -70x and 21 is -7. Therefore, we write -7 to the left of the top row. You can see it here in the color green:

-7	-70x	21
9x	90x²	-27x
	10x	-3

Next, let’s look at the bottom row. We have the terms 90x² and -27x. The greatest common factor of 90x² and -27x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

-7	-70x	21
9x	90x²	-27x
	10x	-3

To find the values below the table, we first divide 90x² by 9x (labeled in blue). This gives us 10x.

90x² ÷ 9x = 10x

You can see this value colored in orange below:

-7	-70x	21
9x	90x²	-27x
	10x	-3

Next, we divide -27x by 9x (labeled in blue). This gives us -3.

-27x ÷ 9x = -3

You can see this value colored in purple below:

-7	-70x	21
9x	90x²	-27x
	10x	-3

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 90x² - 97x + 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:

(9x - 7)(10x - 3)

That’s it! Now you know how to factor the equation 90x² - 97x + 21.

Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor 90x² - 97x + 26
Here is the next quadratic function on our list that we have factored for you.