Factor 60x² + 66x + 18

Here we will show you how to factor the quadratic function 60x² + 66x + 18 using the box method. In other words, we will show you how to factor 60x squared plus 66x plus 18 (60x^2 + 66x + 18) 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 60x² + 66x + 18, like this:

a = 60
b = 66
c = 18

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

6	30x	18
12x	60x²	36x
	5x	3

We put 60x² (a) in the bottom left square and 18 (c) in the top right square, like this:

6	30x	18
12x	60x²	36x
	5x	3

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 60 times 18 (a × c), and add together to equal 66 (b).

More specifically, 60 times 18 is 1080. Therefore, we need to find the two numbers that multiply to equal 1080, and add to equal 66.

? × ? = 1080
? + ? = 66

After looking at this problem, we can see that the two numbers that multiply together to equal 1080, and add together to equal 66, are 30 and 36, as illustrated here:

30 × 36 = 1080
30 + 36 = 66

Now, we can fill in the last two squares in our box with 30x and 36x. Place 30x in the upper left square, and place 36x in the lower right square.

6	30x	18
12x	60x²	36x
	5x	3

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 30x and 18. The greatest common factor of 30x and 18 is 6. Therefore, we write 6 to the left of the top row. You can see it here in the color green:

6	30x	18
12x	60x²	36x
	5x	3

Next, let’s look at the bottom row. We have the terms 60x² and 36x. The greatest common factor of 60x² and 36x is 12x. Therefore, we write 12x to the left of the bottom row. You can see it here in the color blue:

6	30x	18
12x	60x²	36x
	5x	3

To find the values below the table, we first divide 60x² by 12x (labeled in blue). This gives us 5x.

60x² ÷ 12x = 5x

You can see this value colored in orange below:

6	30x	18
12x	60x²	36x
	5x	3

Next, we divide 36x by 12x (labeled in blue). This gives us 3.

36x ÷ 12x = 3

You can see this value colored in purple below:

6	30x	18
12x	60x²	36x
	5x	3

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 60x² + 66x + 18. 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 + 6)(5x + 3)

That’s it! Now you know how to factor the equation 60x² + 66x + 18.

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