Factor -24x² - 50x

Factor -24x² - 50x - 24

Here we will show you how to factor the quadratic function -24x² - 50x - 24 using the box method. In other words, we will show you how to factor negative 24x squared minus 50x minus 24 (-24x^2 - 50x - 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 24x² + 50x + 24. Now we can label the different parts of our equation, like this:

a = 24
b = 50
c = 24

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

6	18x	24
8x	24x²	32x
	3x	4

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

6	18x	24
8x	24x²	32x
	3x	4

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 24 times 24 (a × c), and add together to equal 50 (b).

More specifically, 24 times 24 is 576. Therefore, we need to find the two numbers that multiply to equal 576, and add to equal 50.

? × ? = 576
? + ? = 50

After looking at this problem, we can see that the two numbers that multiply together to equal 576, and add together to equal 50, are 18 and 32, as illustrated here:

18 × 32 = 576
18 + 32 = 50

Now, we can fill in the last two squares in our box with 18x and 32x. Place 18x in the upper left square, and place 32x in the lower right square.

6	18x	24
8x	24x²	32x
	3x	4

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 18x and 24. The greatest common factor of 18x and 24 is 6. Therefore, we write 6 to the left of the top row. You can see it here in the color green:

6	18x	24
8x	24x²	32x
	3x	4

Next, let’s look at the bottom row. We have the terms 24x² and 32x. The greatest common factor of 24x² and 32x is 8x. Therefore, we write 8x to the left of the bottom row. You can see it here in the color blue:

6	18x	24
8x	24x²	32x
	3x	4

To find the values below the table, we first divide 24x² by 8x (labeled in blue). This gives us 3x.

24x² ÷ 8x = 3x

You can see this value colored in orange below:

6	18x	24
8x	24x²	32x
	3x	4

Next, we divide 32x by 8x (labeled in blue). This gives us 4.

32x ÷ 8x = 4

You can see this value colored in purple below:

6	18x	24
8x	24x²	32x
	3x	4

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 -24x² - 50x - 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:

(8x + 6)(3x + 4)

In our original quadratic equation, -24x² - 50x - 24, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(8x + 6)(3x + 4)

That’s it! Now you know how to factor the equation -24x² - 50x - 24.

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