Factor 25x² + 65x + 12

Here we will show you how to factor the quadratic function 25x² + 65x + 12 using the box method. In other words, we will show you how to factor 25x squared plus 65x plus 12 (25x^2 + 65x + 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 25x² + 65x + 12, like this:

a = 25
b = 65
c = 12

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

1	5x	12
5x	25x²	60x
	5x	12

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

1	5x	12
5x	25x²	60x
	5x	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 25 times 12 (a × c), and add together to equal 65 (b).

More specifically, 25 times 12 is 300. Therefore, we need to find the two numbers that multiply to equal 300, and add to equal 65.

? × ? = 300
? + ? = 65

After looking at this problem, we can see that the two numbers that multiply together to equal 300, and add together to equal 65, are 5 and 60, as illustrated here:

5 × 60 = 300
5 + 60 = 65

Now, we can fill in the last two squares in our box with 5x and 60x. Place 5x in the upper left square, and place 60x in the lower right square.

1	5x	12
5x	25x²	60x
	5x	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 5x and 12. The greatest common factor of 5x 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	5x	12
5x	25x²	60x
	5x	12

Next, let’s look at the bottom row. We have the terms 25x² and 60x. The greatest common factor of 25x² and 60x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

1	5x	12
5x	25x²	60x
	5x	12

To find the values below the table, we first divide 25x² by 5x (labeled in blue). This gives us 5x.

25x² ÷ 5x = 5x

You can see this value colored in orange below:

1	5x	12
5x	25x²	60x
	5x	12

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

60x ÷ 5x = 12

You can see this value colored in purple below:

1	5x	12
5x	25x²	60x
	5x	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 25x² + 65x + 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:

(5x + 1)(5x + 12)

That’s it! Now you know how to factor the equation 25x² + 65x + 12.

Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor 25x² + 65x + 22
Here is the next quadratic function on our list that we have factored for you.