Factor 32x² - 97x + 66

Here we will show you how to factor the quadratic function 32x² - 97x + 66 using the box method. In other words, we will show you how to factor 32x squared minus 97x plus 66 (32x^2 - 97x + 66) 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 32x² - 97x + 66, like this:

a = 32
b = -97
c = 66

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

-2	-64x	66
x	32x²	-33x
	32x	-33

We put 32x² (a) in the bottom left square and 66 (c) in the top right square, like this:

-2	-64x	66
x	32x²	-33x
	32x	-33

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 32 times 66 (a × c), and add together to equal -97 (b).

More specifically, 32 times 66 is 2112. Therefore, we need to find the two numbers that multiply to equal 2112, and add to equal -97.

? × ? = 2112
? + ? = -97

After looking at this problem, we can see that the two numbers that multiply together to equal 2112, and add together to equal -97, are -64 and -33, as illustrated here:

-64 × -33 = 2112
-64 + -33 = -97

Now, we can fill in the last two squares in our box with -64x and -33x. Place -64x in the upper left square, and place -33x in the lower right square.

-2	-64x	66
x	32x²	-33x
	32x	-33

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 -64x and 66. The greatest common factor of -64x and 66 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-64x	66
x	32x²	-33x
	32x	-33

Next, let’s look at the bottom row. We have the terms 32x² and -33x. The greatest common factor of 32x² and -33x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-2	-64x	66
x	32x²	-33x
	32x	-33

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

32x² ÷ x = 32x

You can see this value colored in orange below:

-2	-64x	66
x	32x²	-33x
	32x	-33

Next, we divide -33x by x (labeled in blue). This gives us -33.

-33x ÷ x = -33

You can see this value colored in purple below:

-2	-64x	66
x	32x²	-33x
	32x	-33

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 32x² - 97x + 66. 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 - 2)(32x - 33)

That’s it! Now you know how to factor the equation 32x² - 97x + 66.

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