Factor 19x² + 56x + 36

Here we will show you how to factor the quadratic function 19x² + 56x + 36 using the box method. In other words, we will show you how to factor 19x squared plus 56x plus 36 (19x^2 + 56x + 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 19x² + 56x + 36, like this:

a = 19
b = 56
c = 36

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

18	18x	36
19x	19x²	38x
	x	2

We put 19x² (a) in the bottom left square and 36 (c) in the top right square, like this:

18	18x	36
19x	19x²	38x
	x	2

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 19 times 36 (a × c), and add together to equal 56 (b).

More specifically, 19 times 36 is 684. Therefore, we need to find the two numbers that multiply to equal 684, and add to equal 56.

? × ? = 684
? + ? = 56

After looking at this problem, we can see that the two numbers that multiply together to equal 684, and add together to equal 56, are 18 and 38, as illustrated here:

18 × 38 = 684
18 + 38 = 56

Now, we can fill in the last two squares in our box with 18x and 38x. Place 18x in the upper left square, and place 38x in the lower right square.

18	18x	36
19x	19x²	38x
	x	2

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

18	18x	36
19x	19x²	38x
	x	2

Next, let’s look at the bottom row. We have the terms 19x² and 38x. The greatest common factor of 19x² and 38x is 19x. Therefore, we write 19x to the left of the bottom row. You can see it here in the color blue:

18	18x	36
19x	19x²	38x
	x	2

To find the values below the table, we first divide 19x² by 19x (labeled in blue). This gives us x.

19x² ÷ 19x = x

You can see this value colored in orange below:

18	18x	36
19x	19x²	38x
	x	2

Next, we divide 38x by 19x (labeled in blue). This gives us 2.

38x ÷ 19x = 2

You can see this value colored in purple below:

18	18x	36
19x	19x²	38x
	x	2

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 19x² + 56x + 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:

(19x + 18)(x + 2)

That’s it! Now you know how to factor the equation 19x² + 56x + 36.

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