Factor -91x² - 97x

Factor -91x² - 97x - 12

Here we will show you how to factor the quadratic function -91x² - 97x - 12 using the box method. In other words, we will show you how to factor negative 91x squared minus 97x minus 12 (-91x^2 - 97x - 12) 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 + 97x + 12. Now we can label the different parts of our equation, like this:

a = 91
b = 97
c = 12

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

1	13x	12
7x	91x²	84x
	13x	12

We put 91x² (a) in the bottom left square and 12 (c) in the top right square, like this:

1	13x	12
7x	91x²	84x
	13x	12

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 12 (a × c), and add together to equal 97 (b).

More specifically, 91 times 12 is 1092. Therefore, we need to find the two numbers that multiply to equal 1092, and add to equal 97.

? × ? = 1092
? + ? = 97

After looking at this problem, we can see that the two numbers that multiply together to equal 1092, and add together to equal 97, are 13 and 84, as illustrated here:

13 × 84 = 1092
13 + 84 = 97

Now, we can fill in the last two squares in our box with 13x and 84x. Place 13x in the upper left square, and place 84x in the lower right square.

1	13x	12
7x	91x²	84x
	13x	12

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 13x and 12. The greatest common factor of 13x and 12 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	13x	12
7x	91x²	84x
	13x	12

Next, let’s look at the bottom row. We have the terms 91x² and 84x. The greatest common factor of 91x² and 84x is 7x. Therefore, we write 7x to the left of the bottom row. You can see it here in the color blue:

1	13x	12
7x	91x²	84x
	13x	12

To find the values below the table, we first divide 91x² by 7x (labeled in blue). This gives us 13x.

91x² ÷ 7x = 13x

You can see this value colored in orange below:

1	13x	12
7x	91x²	84x
	13x	12

Next, we divide 84x by 7x (labeled in blue). This gives us 12.

84x ÷ 7x = 12

You can see this value colored in purple below:

1	13x	12
7x	91x²	84x
	13x	12

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 - 97x - 12. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(7x + 1)(13x + 12)

In our original quadratic equation, -91x² - 97x - 12, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(7x + 1)(13x + 12)

That’s it! Now you know how to factor the equation -91x² - 97x - 12.

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