Factor 39x² - 85x

Factor 39x² - 85x - 96

Here we will show you how to factor the quadratic function 39x² - 85x - 96 using the box method. In other words, we will show you how to factor 39x squared minus 85x minus 96 (39x^2 - 85x - 96) 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 39x² - 85x - 96, like this:

a = 39
b = -85
c = -96

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

-3	-117x	-96
x	39x²	32x
	39x	32

We put 39x² (a) in the bottom left square and -96 (c) in the top right square, like this:

-3	-117x	-96
x	39x²	32x
	39x	32

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 39 times -96 (a × c), and add together to equal -85 (b).

More specifically, 39 times -96 is -3744. Therefore, we need to find the two numbers that multiply to equal -3744, and add to equal -85.

? × ? = -3744
? + ? = -85

After looking at this problem, we can see that the two numbers that multiply together to equal -3744, and add together to equal -85, are -117 and 32, as illustrated here:

-117 × 32 = -3744
-117 + 32 = -85

Now, we can fill in the last two squares in our box with -117x and 32x. Place -117x in the upper left square, and place 32x in the lower right square.

-3	-117x	-96
x	39x²	32x
	39x	32

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 -117x and -96. The greatest common factor of -117x and -96 is -3. Therefore, we write -3 to the left of the top row. You can see it here in the color green:

-3	-117x	-96
x	39x²	32x
	39x	32

Next, let’s look at the bottom row. We have the terms 39x² and 32x. The greatest common factor of 39x² and 32x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-3	-117x	-96
x	39x²	32x
	39x	32

To find the values below the table, we first divide 39x² by x (labeled in blue). This gives us 39x.

39x² ÷ x = 39x

You can see this value colored in orange below:

-3	-117x	-96
x	39x²	32x
	39x	32

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

32x ÷ x = 32

You can see this value colored in purple below:

-3	-117x	-96
x	39x²	32x
	39x	32

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 39x² - 85x - 96. 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 - 3)(39x + 32)

That’s it! Now you know how to factor the equation 39x² - 85x - 96.

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