Factor 25x² + 65x + 22

Here we will show you how to factor the quadratic function 25x² + 65x + 22 using the box method. In other words, we will show you how to factor 25x squared plus 65x plus 22 (25x^2 + 65x + 22) 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 + 22, like this:

a = 25
b = 65
c = 22

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

2	10x	22
5x	25x²	55x
	5x	11

We put 25x² (a) in the bottom left square and 22 (c) in the top right square, like this:

2	10x	22
5x	25x²	55x
	5x	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 25 times 22 (a × c), and add together to equal 65 (b).

More specifically, 25 times 22 is 550. Therefore, we need to find the two numbers that multiply to equal 550, and add to equal 65.

? × ? = 550
? + ? = 65

After looking at this problem, we can see that the two numbers that multiply together to equal 550, and add together to equal 65, are 10 and 55, as illustrated here:

10 × 55 = 550
10 + 55 = 65

Now, we can fill in the last two squares in our box with 10x and 55x. Place 10x in the upper left square, and place 55x in the lower right square.

2	10x	22
5x	25x²	55x
	5x	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 10x and 22. The greatest common factor of 10x and 22 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	10x	22
5x	25x²	55x
	5x	11

Next, let’s look at the bottom row. We have the terms 25x² and 55x. The greatest common factor of 25x² and 55x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

2	10x	22
5x	25x²	55x
	5x	11

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:

2	10x	22
5x	25x²	55x
	5x	11

Next, we divide 55x by 5x (labeled in blue). This gives us 11.

55x ÷ 5x = 11

You can see this value colored in purple below:

2	10x	22
5x	25x²	55x
	5x	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 25x² + 65x + 22. 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 + 2)(5x + 11)

That’s it! Now you know how to factor the equation 25x² + 65x + 22.

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

Factor 25x² + 65x + 30
Here is the next quadratic function on our list that we have factored for you.