Factor 5x² - 9x

Factor 5x² - 9x - 80

Here we will show you how to factor the quadratic function 5x² - 9x - 80 using the box method. In other words, we will show you how to factor 5x squared minus 9x minus 80 (5x^2 - 9x - 80) 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 5x² - 9x - 80, like this:

a = 5
b = -9
c = -80

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

-5	-25x	-80
x	5x²	16x
	5x	16

We put 5x² (a) in the bottom left square and -80 (c) in the top right square, like this:

-5	-25x	-80
x	5x²	16x
	5x	16

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 5 times -80 (a × c), and add together to equal -9 (b).

More specifically, 5 times -80 is -400. Therefore, we need to find the two numbers that multiply to equal -400, and add to equal -9.

? × ? = -400
? + ? = -9

After looking at this problem, we can see that the two numbers that multiply together to equal -400, and add together to equal -9, are -25 and 16, as illustrated here:

-25 × 16 = -400
-25 + 16 = -9

Now, we can fill in the last two squares in our box with -25x and 16x. Place -25x in the upper left square, and place 16x in the lower right square.

-5	-25x	-80
x	5x²	16x
	5x	16

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 -25x and -80. The greatest common factor of -25x and -80 is -5. Therefore, we write -5 to the left of the top row. You can see it here in the color green:

-5	-25x	-80
x	5x²	16x
	5x	16

Next, let’s look at the bottom row. We have the terms 5x² and 16x. The greatest common factor of 5x² and 16x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-5	-25x	-80
x	5x²	16x
	5x	16

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

5x² ÷ x = 5x

You can see this value colored in orange below:

-5	-25x	-80
x	5x²	16x
	5x	16

Next, we divide 16x by x (labeled in blue). This gives us 16.

16x ÷ x = 16

You can see this value colored in purple below:

-5	-25x	-80
x	5x²	16x
	5x	16

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 5x² - 9x - 80. 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 - 5)(5x + 16)

That’s it! Now you know how to factor the equation 5x² - 9x - 80.

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