Factor 12x² + 56x + 25

Here we will show you how to factor the quadratic function 12x² + 56x + 25 using the box method. In other words, we will show you how to factor 12x squared plus 56x plus 25 (12x^2 + 56x + 25) 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 + 56x + 25, like this:

a = 12
b = 56
c = 25

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

1	6x	25
2x	12x²	50x
	6x	25

We put 12x² (a) in the bottom left square and 25 (c) in the top right square, like this:

1	6x	25
2x	12x²	50x
	6x	25

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 25 (a × c), and add together to equal 56 (b).

More specifically, 12 times 25 is 300. Therefore, we need to find the two numbers that multiply to equal 300, and add to equal 56.

? × ? = 300
? + ? = 56

After looking at this problem, we can see that the two numbers that multiply together to equal 300, and add together to equal 56, are 6 and 50, as illustrated here:

6 × 50 = 300
6 + 50 = 56

Now, we can fill in the last two squares in our box with 6x and 50x. Place 6x in the upper left square, and place 50x in the lower right square.

1	6x	25
2x	12x²	50x
	6x	25

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 6x and 25. The greatest common factor of 6x and 25 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	6x	25
2x	12x²	50x
	6x	25

Next, let’s look at the bottom row. We have the terms 12x² and 50x. The greatest common factor of 12x² and 50x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	6x	25
2x	12x²	50x
	6x	25

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:

1	6x	25
2x	12x²	50x
	6x	25

Next, we divide 50x by 2x (labeled in blue). This gives us 25.

50x ÷ 2x = 25

You can see this value colored in purple below:

1	6x	25
2x	12x²	50x
	6x	25

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 + 56x + 25. 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:

(2x + 1)(6x + 25)

That’s it! Now you know how to factor the equation 12x² + 56x + 25.

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