Factor -9x² + 91x

Factor -9x² + 91x - 90

Here we will show you how to factor the quadratic function -9x² + 91x - 90 using the box method. In other words, we will show you how to factor negative 9x squared plus 91x minus 90 (-9x^2 + 91x - 90) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 9x² - 91x + 90. Now we can label the different parts of our equation, like this:

a = 9
b = -91
c = 90

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

-9	-81x	90
x	9x²	-10x
	9x	-10

We put 9x² (a) in the bottom left square and 90 (c) in the top right square, like this:

-9	-81x	90
x	9x²	-10x
	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 9 times 90 (a × c), and add together to equal -91 (b).

More specifically, 9 times 90 is 810. Therefore, we need to find the two numbers that multiply to equal 810, and add to equal -91.

? × ? = 810
? + ? = -91

After looking at this problem, we can see that the two numbers that multiply together to equal 810, and add together to equal -91, are -81 and -10, as illustrated here:

-81 × -10 = 810
-81 + -10 = -91

Now, we can fill in the last two squares in our box with -81x and -10x. Place -81x in the upper left square, and place -10x in the lower right square.

-9	-81x	90
x	9x²	-10x
	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 -81x and 90. The greatest common factor of -81x and 90 is -9. Therefore, we write -9 to the left of the top row. You can see it here in the color green:

-9	-81x	90
x	9x²	-10x
	9x	-10

Next, let’s look at the bottom row. We have the terms 9x² and -10x. The greatest common factor of 9x² and -10x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-9	-81x	90
x	9x²	-10x
	9x	-10

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

9x² ÷ x = 9x

You can see this value colored in orange below:

-9	-81x	90
x	9x²	-10x
	9x	-10

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

-10x ÷ x = -10

You can see this value colored in purple below:

-9	-81x	90
x	9x²	-10x
	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 -9x² + 91x - 90. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(x - 9)(9x - 10)

In our original quadratic equation, -9x² + 91x - 90, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

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

That’s it! Now you know how to factor the equation -9x² + 91x - 90.

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