Factor 18x² + 51x + 36

Here we will show you how to factor the quadratic function 18x² + 51x + 36 using the box method. In other words, we will show you how to factor 18x squared plus 51x plus 36 (18x^2 + 51x + 36) 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 + 51x + 36, like this:

a = 18
b = 51
c = 36

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

12	24x	36
9x	18x²	27x
	2x	3

We put 18x² (a) in the bottom left square and 36 (c) in the top right square, like this:

12	24x	36
9x	18x²	27x
	2x	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 18 times 36 (a × c), and add together to equal 51 (b).

More specifically, 18 times 36 is 648. Therefore, we need to find the two numbers that multiply to equal 648, and add to equal 51.

? × ? = 648
? + ? = 51

After looking at this problem, we can see that the two numbers that multiply together to equal 648, and add together to equal 51, are 24 and 27, as illustrated here:

24 × 27 = 648
24 + 27 = 51

Now, we can fill in the last two squares in our box with 24x and 27x. Place 24x in the upper left square, and place 27x in the lower right square.

12	24x	36
9x	18x²	27x
	2x	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 24x and 36. The greatest common factor of 24x and 36 is 12. Therefore, we write 12 to the left of the top row. You can see it here in the color green:

12	24x	36
9x	18x²	27x
	2x	3

Next, let’s look at the bottom row. We have the terms 18x² and 27x. The greatest common factor of 18x² and 27x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

12	24x	36
9x	18x²	27x
	2x	3

To find the values below the table, we first divide 18x² by 9x (labeled in blue). This gives us 2x.

18x² ÷ 9x = 2x

You can see this value colored in orange below:

12	24x	36
9x	18x²	27x
	2x	3

Next, we divide 27x by 9x (labeled in blue). This gives us 3.

27x ÷ 9x = 3

You can see this value colored in purple below:

12	24x	36
9x	18x²	27x
	2x	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 18x² + 51x + 36. 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:

(9x + 12)(2x + 3)

That’s it! Now you know how to factor the equation 18x² + 51x + 36.

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