Factor 12x² + 80x + 100

Here we will show you how to factor the quadratic function 12x² + 80x + 100 using the box method. In other words, we will show you how to factor 12x squared plus 80x plus 100 (12x^2 + 80x + 100) 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 12x² + 80x + 100, like this:

a = 12
b = 80
c = 100

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

20	20x	100
12x	12x²	60x
	x	5

We put 12x² (a) in the bottom left square and 100 (c) in the top right square, like this:

20	20x	100
12x	12x²	60x
	x	5

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 100 (a × c), and add together to equal 80 (b).

More specifically, 12 times 100 is 1200. Therefore, we need to find the two numbers that multiply to equal 1200, and add to equal 80.

? × ? = 1200
? + ? = 80

After looking at this problem, we can see that the two numbers that multiply together to equal 1200, and add together to equal 80, are 20 and 60, as illustrated here:

20 × 60 = 1200
20 + 60 = 80

Now, we can fill in the last two squares in our box with 20x and 60x. Place 20x in the upper left square, and place 60x in the lower right square.

20	20x	100
12x	12x²	60x
	x	5

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 20x and 100. The greatest common factor of 20x and 100 is 20. Therefore, we write 20 to the left of the top row. You can see it here in the color green:

20	20x	100
12x	12x²	60x
	x	5

Next, let’s look at the bottom row. We have the terms 12x² and 60x. The greatest common factor of 12x² and 60x is 12x. Therefore, we write 12x to the left of the bottom row. You can see it here in the color blue:

20	20x	100
12x	12x²	60x
	x	5

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

12x² ÷ 12x = x

You can see this value colored in orange below:

20	20x	100
12x	12x²	60x
	x	5

Next, we divide 60x by 12x (labeled in blue). This gives us 5.

60x ÷ 12x = 5

You can see this value colored in purple below:

20	20x	100
12x	12x²	60x
	x	5

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 + 80x + 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 the answer:

(12x + 20)(x + 5)

That’s it! Now you know how to factor the equation 12x² + 80x + 100.

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