Factor -91x² - 100x

Factor -91x² - 100x - 9

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

a = 91
b = 100
c = 9

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

9	9x	9
91x	91x²	91x
	x	1

We put 91x² (a) in the bottom left square and 9 (c) in the top right square, like this:

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

More specifically, 91 times 9 is 819. Therefore, we need to find the two numbers that multiply to equal 819, and add to equal 100.

? × ? = 819
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal 819, and add together to equal 100, are 9 and 91, as illustrated here:

9 × 91 = 819
9 + 91 = 100

Now, we can fill in the last two squares in our box with 9x and 91x. Place 9x in the upper left square, and place 91x in the lower right square.

9	9x	9
91x	91x²	91x
	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 9x and 9. The greatest common factor of 9x and 9 is 9. Therefore, we write 9 to the left of the top row. You can see it here in the color green:

9	9x	9
91x	91x²	91x
	x	1

Next, let’s look at the bottom row. We have the terms 91x² and 91x. The greatest common factor of 91x² and 91x is 91x. Therefore, we write 91x to the left of the bottom row. You can see it here in the color blue:

9	9x	9
91x	91x²	91x
	x	1

To find the values below the table, we first divide 91x² by 91x (labeled in blue). This gives us x.

91x² ÷ 91x = x

You can see this value colored in orange below:

9	9x	9
91x	91x²	91x
	x	1

Next, we divide 91x by 91x (labeled in blue). This gives us 1.

91x ÷ 91x = 1

You can see this value colored in purple below:

9	9x	9
91x	91x²	91x
	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 -91x² - 100x - 9. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(91x + 9)(x + 1)

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

-(91x + 9)(x + 1)

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

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