Factor 6x² + 44x + 64

Here we will show you how to factor the quadratic function 6x² + 44x + 64 using the box method. In other words, we will show you how to factor 6x squared plus 44x plus 64 (6x^2 + 44x + 64) 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 6x² + 44x + 64, like this:

a = 6
b = 44
c = 64

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

4	12x	64
2x	6x²	32x
	3x	16

We put 6x² (a) in the bottom left square and 64 (c) in the top right square, like this:

4	12x	64
2x	6x²	32x
	3x	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 6 times 64 (a × c), and add together to equal 44 (b).

More specifically, 6 times 64 is 384. Therefore, we need to find the two numbers that multiply to equal 384, and add to equal 44.

? × ? = 384
? + ? = 44

After looking at this problem, we can see that the two numbers that multiply together to equal 384, and add together to equal 44, are 12 and 32, as illustrated here:

12 × 32 = 384
12 + 32 = 44

Now, we can fill in the last two squares in our box with 12x and 32x. Place 12x in the upper left square, and place 32x in the lower right square.

4	12x	64
2x	6x²	32x
	3x	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 12x and 64. The greatest common factor of 12x and 64 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	12x	64
2x	6x²	32x
	3x	16

Next, let’s look at the bottom row. We have the terms 6x² and 32x. The greatest common factor of 6x² and 32x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

4	12x	64
2x	6x²	32x
	3x	16

To find the values below the table, we first divide 6x² by 2x (labeled in blue). This gives us 3x.

6x² ÷ 2x = 3x

You can see this value colored in orange below:

4	12x	64
2x	6x²	32x
	3x	16

Next, we divide 32x by 2x (labeled in blue). This gives us 16.

32x ÷ 2x = 16

You can see this value colored in purple below:

4	12x	64
2x	6x²	32x
	3x	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 6x² + 44x + 64. 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:

(2x + 4)(3x + 16)

That’s it! Now you know how to factor the equation 6x² + 44x + 64.

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