Factor 68x² + 98x + 32

Here we will show you how to factor the quadratic function 68x² + 98x + 32 using the box method. In other words, we will show you how to factor 68x squared plus 98x plus 32 (68x^2 + 98x + 32) 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 68x² + 98x + 32, like this:

a = 68
b = 98
c = 32

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

2	34x	32
4x	68x²	64x
	17x	16

We put 68x² (a) in the bottom left square and 32 (c) in the top right square, like this:

2	34x	32
4x	68x²	64x
	17x	16

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 68 times 32 (a × c), and add together to equal 98 (b).

More specifically, 68 times 32 is 2176. Therefore, we need to find the two numbers that multiply to equal 2176, and add to equal 98.

? × ? = 2176
? + ? = 98

After looking at this problem, we can see that the two numbers that multiply together to equal 2176, and add together to equal 98, are 34 and 64, as illustrated here:

34 × 64 = 2176
34 + 64 = 98

Now, we can fill in the last two squares in our box with 34x and 64x. Place 34x in the upper left square, and place 64x in the lower right square.

2	34x	32
4x	68x²	64x
	17x	16

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 34x and 32. The greatest common factor of 34x and 32 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	34x	32
4x	68x²	64x
	17x	16

Next, let’s look at the bottom row. We have the terms 68x² and 64x. The greatest common factor of 68x² and 64x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

2	34x	32
4x	68x²	64x
	17x	16

To find the values below the table, we first divide 68x² by 4x (labeled in blue). This gives us 17x.

68x² ÷ 4x = 17x

You can see this value colored in orange below:

2	34x	32
4x	68x²	64x
	17x	16

Next, we divide 64x by 4x (labeled in blue). This gives us 16.

64x ÷ 4x = 16

You can see this value colored in purple below:

2	34x	32
4x	68x²	64x
	17x	16

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 68x² + 98x + 32. 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:

(4x + 2)(17x + 16)

That’s it! Now you know how to factor the equation 68x² + 98x + 32.

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