Factor 32x² - 28x

Factor 32x² - 28x - 85

Here we will show you how to factor the quadratic function 32x² - 28x - 85 using the box method. In other words, we will show you how to factor 32x squared minus 28x minus 85 (32x^2 - 28x - 85) 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 - 28x - 85, like this:

a = 32
b = -28
c = -85

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

-17	-68x	-85
8x	32x²	40x
	4x	5

We put 32x² (a) in the bottom left square and -85 (c) in the top right square, like this:

-17	-68x	-85
8x	32x²	40x
	4x	5

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 -85 (a × c), and add together to equal -28 (b).

More specifically, 32 times -85 is -2720. Therefore, we need to find the two numbers that multiply to equal -2720, and add to equal -28.

? × ? = -2720
? + ? = -28

After looking at this problem, we can see that the two numbers that multiply together to equal -2720, and add together to equal -28, are -68 and 40, as illustrated here:

-68 × 40 = -2720
-68 + 40 = -28

Now, we can fill in the last two squares in our box with -68x and 40x. Place -68x in the upper left square, and place 40x in the lower right square.

-17	-68x	-85
8x	32x²	40x
	4x	5

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 -68x and -85. The greatest common factor of -68x and -85 is -17. Therefore, we write -17 to the left of the top row. You can see it here in the color green:

-17	-68x	-85
8x	32x²	40x
	4x	5

Next, let’s look at the bottom row. We have the terms 32x² and 40x. The greatest common factor of 32x² and 40x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

-17	-68x	-85
8x	32x²	40x
	4x	5

To find the values below the table, we first divide 32x² by 8x (labeled in blue). This gives us 4x.

32x² ÷ 8x = 4x

You can see this value colored in orange below:

-17	-68x	-85
8x	32x²	40x
	4x	5

Next, we divide 40x by 8x (labeled in blue). This gives us 5.

40x ÷ 8x = 5

You can see this value colored in purple below:

-17	-68x	-85
8x	32x²	40x
	4x	5

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 - 28x - 85. 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:

(8x - 17)(4x + 5)

That’s it! Now you know how to factor the equation 32x² - 28x - 85.

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