Factor -12x² + 76x

Factor -12x² + 76x - 99

Here we will show you how to factor the quadratic function -12x² + 76x - 99 using the box method. In other words, we will show you how to factor negative 12x squared plus 76x minus 99 (-12x^2 + 76x - 99) 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 12x² - 76x + 99. Now we can label the different parts of our equation, like this:

a = 12
b = -76
c = 99

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

-9	-54x	99
2x	12x²	-22x
	6x	-11

We put 12x² (a) in the bottom left square and 99 (c) in the top right square, like this:

-9	-54x	99
2x	12x²	-22x
	6x	-11

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 12 times 99 (a × c), and add together to equal -76 (b).

More specifically, 12 times 99 is 1188. Therefore, we need to find the two numbers that multiply to equal 1188, and add to equal -76.

? × ? = 1188
? + ? = -76

After looking at this problem, we can see that the two numbers that multiply together to equal 1188, and add together to equal -76, are -54 and -22, as illustrated here:

-54 × -22 = 1188
-54 + -22 = -76

Now, we can fill in the last two squares in our box with -54x and -22x. Place -54x in the upper left square, and place -22x in the lower right square.

-9	-54x	99
2x	12x²	-22x
	6x	-11

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 -54x and 99. The greatest common factor of -54x and 99 is -9. Therefore, we write -9 to the left of the top row. You can see it here in the color green:

-9	-54x	99
2x	12x²	-22x
	6x	-11

Next, let’s look at the bottom row. We have the terms 12x² and -22x. The greatest common factor of 12x² and -22x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

-9	-54x	99
2x	12x²	-22x
	6x	-11

To find the values below the table, we first divide 12x² by 2x (labeled in blue). This gives us 6x.

12x² ÷ 2x = 6x

You can see this value colored in orange below:

-9	-54x	99
2x	12x²	-22x
	6x	-11

Next, we divide -22x by 2x (labeled in blue). This gives us -11.

-22x ÷ 2x = -11

You can see this value colored in purple below:

-9	-54x	99
2x	12x²	-22x
	6x	-11

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 -12x² + 76x - 99. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(2x - 9)(6x - 11)

In our original quadratic equation, -12x² + 76x - 99, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x - 9)(6x - 11)

That’s it! Now you know how to factor the equation -12x² + 76x - 99.

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