Factor 18x² + 39x + 20

Here we will show you how to factor the quadratic function 18x² + 39x + 20 using the box method. In other words, we will show you how to factor 18x squared plus 39x plus 20 (18x^2 + 39x + 20) 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 18x² + 39x + 20, like this:

a = 18
b = 39
c = 20

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

5	15x	20
6x	18x²	24x
	3x	4

We put 18x² (a) in the bottom left square and 20 (c) in the top right square, like this:

5	15x	20
6x	18x²	24x
	3x	4

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 18 times 20 (a × c), and add together to equal 39 (b).

More specifically, 18 times 20 is 360. Therefore, we need to find the two numbers that multiply to equal 360, and add to equal 39.

? × ? = 360
? + ? = 39

After looking at this problem, we can see that the two numbers that multiply together to equal 360, and add together to equal 39, are 15 and 24, as illustrated here:

15 × 24 = 360
15 + 24 = 39

Now, we can fill in the last two squares in our box with 15x and 24x. Place 15x in the upper left square, and place 24x in the lower right square.

5	15x	20
6x	18x²	24x
	3x	4

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 15x and 20. The greatest common factor of 15x and 20 is 5. Therefore, we write 5 to the left of the top row. You can see it here in the color green:

5	15x	20
6x	18x²	24x
	3x	4

Next, let’s look at the bottom row. We have the terms 18x² and 24x. The greatest common factor of 18x² and 24x is 6x. Therefore, we write 6x to the left of the bottom row. You can see it here in the color blue:

5	15x	20
6x	18x²	24x
	3x	4

To find the values below the table, we first divide 18x² by 6x (labeled in blue). This gives us 3x.

18x² ÷ 6x = 3x

You can see this value colored in orange below:

5	15x	20
6x	18x²	24x
	3x	4

Next, we divide 24x by 6x (labeled in blue). This gives us 4.

24x ÷ 6x = 4

You can see this value colored in purple below:

5	15x	20
6x	18x²	24x
	3x	4

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 18x² + 39x + 20. 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:

(6x + 5)(3x + 4)

That’s it! Now you know how to factor the equation 18x² + 39x + 20.

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