Factor 40x² + 39x

Factor 40x² + 39x - 22

Here we will show you how to factor the quadratic function 40x² + 39x - 22 using the box method. In other words, we will show you how to factor 40x squared plus 39x minus 22 (40x^2 + 39x - 22) 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 40x² + 39x - 22, like this:

a = 40
b = 39
c = -22

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

-2	-16x	-22
5x	40x²	55x
	8x	11

We put 40x² (a) in the bottom left square and -22 (c) in the top right square, like this:

-2	-16x	-22
5x	40x²	55x
	8x	11

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 40 times -22 (a × c), and add together to equal 39 (b).

More specifically, 40 times -22 is -880. Therefore, we need to find the two numbers that multiply to equal -880, and add to equal 39.

? × ? = -880
? + ? = 39

After looking at this problem, we can see that the two numbers that multiply together to equal -880, and add together to equal 39, are -16 and 55, as illustrated here:

-16 × 55 = -880
-16 + 55 = 39

Now, we can fill in the last two squares in our box with -16x and 55x. Place -16x in the upper left square, and place 55x in the lower right square.

-2	-16x	-22
5x	40x²	55x
	8x	11

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 -22. The greatest common factor of -16x and -22 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-16x	-22
5x	40x²	55x
	8x	11

Next, let’s look at the bottom row. We have the terms 40x² and 55x. The greatest common factor of 40x² and 55x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

-2	-16x	-22
5x	40x²	55x
	8x	11

To find the values below the table, we first divide 40x² by 5x (labeled in blue). This gives us 8x.

40x² ÷ 5x = 8x

You can see this value colored in orange below:

-2	-16x	-22
5x	40x²	55x
	8x	11

Next, we divide 55x by 5x (labeled in blue). This gives us 11.

55x ÷ 5x = 11

You can see this value colored in purple below:

-2	-16x	-22
5x	40x²	55x
	8x	11

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 40x² + 39x - 22. 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:

(5x - 2)(8x + 11)

That’s it! Now you know how to factor the equation 40x² + 39x - 22.

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