Factor 64x² + 64x + 16

Here we will show you how to factor the quadratic function 64x² + 64x + 16 using the box method. In other words, we will show you how to factor 64x squared plus 64x plus 16 (64x^2 + 64x + 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 64x² + 64x + 16, like this:

a = 64
b = 64
c = 16

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

16	32x	16
32x	64x²	32x
	2x	1

We put 64x² (a) in the bottom left square and 16 (c) in the top right square, like this:

16	32x	16
32x	64x²	32x
	2x	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 64 times 16 (a × c), and add together to equal 64 (b).

More specifically, 64 times 16 is 1024. Therefore, we need to find the two numbers that multiply to equal 1024, and add to equal 64.

? × ? = 1024
? + ? = 64

After looking at this problem, we can see that the two numbers that multiply together to equal 1024, and add together to equal 64, are 32 and 32, as illustrated here:

32 × 32 = 1024
32 + 32 = 64

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

16	32x	16
32x	64x²	32x
	2x	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 32x and 16. The greatest common factor of 32x 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	32x	16
32x	64x²	32x
	2x	1

Next, let’s look at the bottom row. We have the terms 64x² and 32x. The greatest common factor of 64x² and 32x is 32x. Therefore, we write 32x to the left of the bottom row. You can see it here in the color blue:

16	32x	16
32x	64x²	32x
	2x	1

To find the values below the table, we first divide 64x² by 32x (labeled in blue). This gives us 2x.

64x² ÷ 32x = 2x

You can see this value colored in orange below:

16	32x	16
32x	64x²	32x
	2x	1

Next, we divide 32x by 32x (labeled in blue). This gives us 1.

32x ÷ 32x = 1

You can see this value colored in purple below:

16	32x	16
32x	64x²	32x
	2x	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 64x² + 64x + 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:

(32x + 16)(2x + 1)

That’s it! Now you know how to factor the equation 64x² + 64x + 16.

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