Factor 22x² + 38x + 16

Here we will show you how to factor the quadratic function 22x² + 38x + 16 using the box method. In other words, we will show you how to factor 22x squared plus 38x plus 16 (22x^2 + 38x + 16) 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 22x² + 38x + 16, like this:

a = 22
b = 38
c = 16

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

16	16x	16
22x	22x²	22x
	x	1

We put 22x² (a) in the bottom left square and 16 (c) in the top right square, like this:

16	16x	16
22x	22x²	22x
	x	1

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 22 times 16 (a × c), and add together to equal 38 (b).

More specifically, 22 times 16 is 352. Therefore, we need to find the two numbers that multiply to equal 352, and add to equal 38.

? × ? = 352
? + ? = 38

After looking at this problem, we can see that the two numbers that multiply together to equal 352, and add together to equal 38, are 16 and 22, as illustrated here:

16 × 22 = 352
16 + 22 = 38

Now, we can fill in the last two squares in our box with 16x and 22x. Place 16x in the upper left square, and place 22x in the lower right square.

16	16x	16
22x	22x²	22x
	x	1

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 16x and 16. The greatest common factor of 16x and 16 is 16. Therefore, we write 16 to the left of the top row. You can see it here in the color green:

16	16x	16
22x	22x²	22x
	x	1

Next, let’s look at the bottom row. We have the terms 22x² and 22x. The greatest common factor of 22x² and 22x is 22x. Therefore, we write 22x to the left of the bottom row. You can see it here in the color blue:

16	16x	16
22x	22x²	22x
	x	1

To find the values below the table, we first divide 22x² by 22x (labeled in blue). This gives us x.

22x² ÷ 22x = x

You can see this value colored in orange below:

16	16x	16
22x	22x²	22x
	x	1

Next, we divide 22x by 22x (labeled in blue). This gives us 1.

22x ÷ 22x = 1

You can see this value colored in purple below:

16	16x	16
22x	22x²	22x
	x	1

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 22x² + 38x + 16. 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:

(22x + 16)(x + 1)

That’s it! Now you know how to factor the equation 22x² + 38x + 16.

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