Factor 15x² + 61x + 34

Here we will show you how to factor the quadratic function 15x² + 61x + 34 using the box method. In other words, we will show you how to factor 15x squared plus 61x plus 34 (15x^2 + 61x + 34) 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 15x² + 61x + 34, like this:

a = 15
b = 61
c = 34

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

2	10x	34
3x	15x²	51x
	5x	17

We put 15x² (a) in the bottom left square and 34 (c) in the top right square, like this:

2	10x	34
3x	15x²	51x
	5x	17

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 15 times 34 (a × c), and add together to equal 61 (b).

More specifically, 15 times 34 is 510. Therefore, we need to find the two numbers that multiply to equal 510, and add to equal 61.

? × ? = 510
? + ? = 61

After looking at this problem, we can see that the two numbers that multiply together to equal 510, and add together to equal 61, are 10 and 51, as illustrated here:

10 × 51 = 510
10 + 51 = 61

Now, we can fill in the last two squares in our box with 10x and 51x. Place 10x in the upper left square, and place 51x in the lower right square.

2	10x	34
3x	15x²	51x
	5x	17

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 34. The greatest common factor of 10x and 34 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	10x	34
3x	15x²	51x
	5x	17

Next, let’s look at the bottom row. We have the terms 15x² and 51x. The greatest common factor of 15x² and 51x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

2	10x	34
3x	15x²	51x
	5x	17

To find the values below the table, we first divide 15x² by 3x (labeled in blue). This gives us 5x.

15x² ÷ 3x = 5x

You can see this value colored in orange below:

2	10x	34
3x	15x²	51x
	5x	17

Next, we divide 51x by 3x (labeled in blue). This gives us 17.

51x ÷ 3x = 17

You can see this value colored in purple below:

2	10x	34
3x	15x²	51x
	5x	17

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 15x² + 61x + 34. 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:

(3x + 2)(5x + 17)

That’s it! Now you know how to factor the equation 15x² + 61x + 34.

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

Factor 15x² + 61x + 46
Here is the next quadratic function on our list that we have factored for you.