Factor -100x² - 100x

Factor -100x² - 100x - 24

Here we will show you how to factor the quadratic function -100x² - 100x - 24 using the box method. In other words, we will show you how to factor negative 100x squared minus 100x minus 24 (-100x^2 - 100x - 24) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 100x² + 100x + 24. Now we can label the different parts of our equation, like this:

a = 100
b = 100
c = 24

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

8	40x	24
20x	100x²	60x
	5x	3

We put 100x² (a) in the bottom left square and 24 (c) in the top right square, like this:

8	40x	24
20x	100x²	60x
	5x	3

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 100 times 24 (a × c), and add together to equal 100 (b).

More specifically, 100 times 24 is 2400. Therefore, we need to find the two numbers that multiply to equal 2400, and add to equal 100.

? × ? = 2400
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal 2400, and add together to equal 100, are 40 and 60, as illustrated here:

40 × 60 = 2400
40 + 60 = 100

Now, we can fill in the last two squares in our box with 40x and 60x. Place 40x in the upper left square, and place 60x in the lower right square.

8	40x	24
20x	100x²	60x
	5x	3

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 40x and 24. The greatest common factor of 40x and 24 is 8. Therefore, we write 8 to the left of the top row. You can see it here in the color green:

8	40x	24
20x	100x²	60x
	5x	3

Next, let’s look at the bottom row. We have the terms 100x² and 60x. The greatest common factor of 100x² and 60x is 20x. Therefore, we write 20x to the left of the bottom row. You can see it here in the color blue:

8	40x	24
20x	100x²	60x
	5x	3

To find the values below the table, we first divide 100x² by 20x (labeled in blue). This gives us 5x.

100x² ÷ 20x = 5x

You can see this value colored in orange below:

8	40x	24
20x	100x²	60x
	5x	3

Next, we divide 60x by 20x (labeled in blue). This gives us 3.

60x ÷ 20x = 3

You can see this value colored in purple below:

8	40x	24
20x	100x²	60x
	5x	3

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 -100x² - 100x - 24. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(20x + 8)(5x + 3)

In our original quadratic equation, -100x² - 100x - 24, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(20x + 8)(5x + 3)

That’s it! Now you know how to factor the equation -100x² - 100x - 24.

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