Factor -9x² + 91x

Factor -9x² + 91x - 10

Here we will show you how to factor the quadratic function -9x² + 91x - 10 using the box method. In other words, we will show you how to factor negative 9x squared plus 91x minus 10 (-9x^2 + 91x - 10) 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 + 10. Now we can label the different parts of our equation, like this:

a = 9
b = -91
c = 10

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

-10	-90x	10
x	9x²	-x
	9x	-1

We put 9x² (a) in the bottom left square and 10 (c) in the top right square, like this:

-10	-90x	10
x	9x²	-x
	9x	-1

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 10 (a × c), and add together to equal -91 (b).

More specifically, 9 times 10 is 90. Therefore, we need to find the two numbers that multiply to equal 90, and add to equal -91.

? × ? = 90
? + ? = -91

After looking at this problem, we can see that the two numbers that multiply together to equal 90, and add together to equal -91, are -90 and -1, as illustrated here:

-90 × -1 = 90
-90 + -1 = -91

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

-10	-90x	10
x	9x²	-x
	9x	-1

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

-10	-90x	10
x	9x²	-x
	9x	-1

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

-10	-90x	10
x	9x²	-x
	9x	-1

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:

-10	-90x	10
x	9x²	-x
	9x	-1

Next, we divide -x by x (labeled in blue). This gives us -1.

-x ÷ x = -1

You can see this value colored in purple below:

-10	-90x	10
x	9x²	-x
	9x	-1

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 - 10. 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 - 10)(9x - 1)

In our original quadratic equation, -9x² + 91x - 10, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 10)(9x - 1)

That’s it! Now you know how to factor the equation -9x² + 91x - 10.

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