Factor -9x² + 91x + 100

Here we will show you how to factor the quadratic function -9x² + 91x + 100 using the box method. In other words, we will show you how to factor negative 9x squared plus 91x plus 100 (-9x^2 + 91x + 100) 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 - 100. Now we can label the different parts of our equation, like this:

a = 9
b = -91
c = -100

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

-100	-100x	-100
9x	9x²	9x
	x	1

We put 9x² (a) in the bottom left square and -100 (c) in the top right square, like this:

-100	-100x	-100
9x	9x²	9x
	x	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 -100 (a × c), and add together to equal -91 (b).

More specifically, 9 times -100 is -900. Therefore, we need to find the two numbers that multiply to equal -900, and add to equal -91.

? × ? = -900
? + ? = -91

After looking at this problem, we can see that the two numbers that multiply together to equal -900, and add together to equal -91, are -100 and 9, as illustrated here:

-100 × 9 = -900
-100 + 9 = -91

Now, we can fill in the last two squares in our box with -100x and 9x. Place -100x in the upper left square, and place 9x in the lower right square.

-100	-100x	-100
9x	9x²	9x
	x	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 -100x and -100. The greatest common factor of -100x and -100 is -100. Therefore, we write -100 to the left of the top row. You can see it here in the color green:

-100	-100x	-100
9x	9x²	9x
	x	1

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

-100	-100x	-100
9x	9x²	9x
	x	1

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

9x² ÷ 9x = x

You can see this value colored in orange below:

-100	-100x	-100
9x	9x²	9x
	x	1

Next, we divide 9x by 9x (labeled in blue). This gives us 1.

9x ÷ 9x = 1

You can see this value colored in purple below:

-100	-100x	-100
9x	9x²	9x
	x	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 + 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:

(9x - 100)(x + 1)

In our original quadratic equation, -9x² + 91x + 100, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(9x - 100)(x + 1)

That’s it! Now you know how to factor the equation -9x² + 91x + 100.

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Factor -9x² + 92x - 99
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