Factor x² - 18x + 32

Here we will show you how to factor the quadratic function x² - 18x + 32 using the box method. In other words, we will show you how to factor x squared minus 18x plus 32 (x^2 - 18x + 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 x² - 18x + 32, like this:

a = 1
b = -18
c = 32

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

-16	-16x	32
x	x²	-2x
	x	-2

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

-16	-16x	32
x	x²	-2x
	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 1 times 32 (a × c), and add together to equal -18 (b).

More specifically, 1 times 32 is 32. Therefore, we need to find the two numbers that multiply to equal 32, and add to equal -18.

? × ? = 32
? + ? = -18

After looking at this problem, we can see that the two numbers that multiply together to equal 32, and add together to equal -18, are -16 and -2, as illustrated here:

-16 × -2 = 32
-16 + -2 = -18

Now, we can fill in the last two squares in our box with -16x and -2x. Place -16x in the upper left square, and place -2x in the lower right square.

-16	-16x	32
x	x²	-2x
	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 -16x and 32. The greatest common factor of -16x and 32 is -16. Therefore, we write -16 to the left of the top row. You can see it here in the color green:

-16	-16x	32
x	x²	-2x
	x	-2

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

-16	-16x	32
x	x²	-2x
	x	-2

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

x² ÷ x = x

You can see this value colored in orange below:

-16	-16x	32
x	x²	-2x
	x	-2

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

-2x ÷ x = -2

You can see this value colored in purple below:

-16	-16x	32
x	x²	-2x
	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 x² - 18x + 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:

(x - 16)(x - 2)

That’s it! Now you know how to factor the equation x² - 18x + 32.

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