Factor 99x² + 84x + 12

Here we will show you how to factor the quadratic function 99x² + 84x + 12 using the box method. In other words, we will show you how to factor 99x squared plus 84x plus 12 (99x^2 + 84x + 12) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 99x² + 84x + 12, like this:

a = 99
b = 84
c = 12

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

6	18x	12
33x	99x²	66x
	3x	2

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

6	18x	12
33x	99x²	66x
	3x	2

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 99 times 12 (a × c), and add together to equal 84 (b).

More specifically, 99 times 12 is 1188. Therefore, we need to find the two numbers that multiply to equal 1188, and add to equal 84.

? × ? = 1188
? + ? = 84

After looking at this problem, we can see that the two numbers that multiply together to equal 1188, and add together to equal 84, are 18 and 66, as illustrated here:

18 × 66 = 1188
18 + 66 = 84

Now, we can fill in the last two squares in our box with 18x and 66x. Place 18x in the upper left square, and place 66x in the lower right square.

6	18x	12
33x	99x²	66x
	3x	2

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 18x and 12. The greatest common factor of 18x and 12 is 6. Therefore, we write 6 to the left of the top row. You can see it here in the color green:

6	18x	12
33x	99x²	66x
	3x	2

Next, let’s look at the bottom row. We have the terms 99x² and 66x. The greatest common factor of 99x² and 66x is 33x. Therefore, we write 33x to the left of the bottom row. You can see it here in the color blue:

6	18x	12
33x	99x²	66x
	3x	2

To find the values below the table, we first divide 99x² by 33x (labeled in blue). This gives us 3x.

99x² ÷ 33x = 3x

You can see this value colored in orange below:

6	18x	12
33x	99x²	66x
	3x	2

Next, we divide 66x by 33x (labeled in blue). This gives us 2.

66x ÷ 33x = 2

You can see this value colored in purple below:

6	18x	12
33x	99x²	66x
	3x	2

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 99x² + 84x + 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 the answer:

(33x + 6)(3x + 2)

That’s it! Now you know how to factor the equation 99x² + 84x + 12.

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