Factor -24x² - 98x

Factor -24x² - 98x - 93

Here we will show you how to factor the quadratic function -24x² - 98x - 93 using the box method. In other words, we will show you how to factor negative 24x squared minus 98x minus 93 (-24x^2 - 98x - 93) 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 + 98x + 93. Now we can label the different parts of our equation, like this:

a = 24
b = 98
c = 93

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

3	36x	93
2x	24x²	62x
	12x	31

We put 24x² (a) in the bottom left square and 93 (c) in the top right square, like this:

3	36x	93
2x	24x²	62x
	12x	31

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 93 (a × c), and add together to equal 98 (b).

More specifically, 24 times 93 is 2232. Therefore, we need to find the two numbers that multiply to equal 2232, and add to equal 98.

? × ? = 2232
? + ? = 98

After looking at this problem, we can see that the two numbers that multiply together to equal 2232, and add together to equal 98, are 36 and 62, as illustrated here:

36 × 62 = 2232
36 + 62 = 98

Now, we can fill in the last two squares in our box with 36x and 62x. Place 36x in the upper left square, and place 62x in the lower right square.

3	36x	93
2x	24x²	62x
	12x	31

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 36x and 93. The greatest common factor of 36x and 93 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	36x	93
2x	24x²	62x
	12x	31

Next, let’s look at the bottom row. We have the terms 24x² and 62x. The greatest common factor of 24x² and 62x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

3	36x	93
2x	24x²	62x
	12x	31

To find the values below the table, we first divide 24x² by 2x (labeled in blue). This gives us 12x.

24x² ÷ 2x = 12x

You can see this value colored in orange below:

3	36x	93
2x	24x²	62x
	12x	31

Next, we divide 62x by 2x (labeled in blue). This gives us 31.

62x ÷ 2x = 31

You can see this value colored in purple below:

3	36x	93
2x	24x²	62x
	12x	31

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 - 98x - 93. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(2x + 3)(12x + 31)

In our original quadratic equation, -24x² - 98x - 93, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 3)(12x + 31)

That’s it! Now you know how to factor the equation -24x² - 98x - 93.

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