Factor -48x² - 99x

Factor -48x² - 99x - 51

Here we will show you how to factor the quadratic function -48x² - 99x - 51 using the box method. In other words, we will show you how to factor negative 48x squared minus 99x minus 51 (-48x^2 - 99x - 51) 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 48x² + 99x + 51. Now we can label the different parts of our equation, like this:

a = 48
b = 99
c = 51

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

3	48x	51
3x	48x²	51x
	16x	17

We put 48x² (a) in the bottom left square and 51 (c) in the top right square, like this:

3	48x	51
3x	48x²	51x
	16x	17

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 48 times 51 (a × c), and add together to equal 99 (b).

More specifically, 48 times 51 is 2448. Therefore, we need to find the two numbers that multiply to equal 2448, and add to equal 99.

? × ? = 2448
? + ? = 99

After looking at this problem, we can see that the two numbers that multiply together to equal 2448, and add together to equal 99, are 48 and 51, as illustrated here:

48 × 51 = 2448
48 + 51 = 99

Now, we can fill in the last two squares in our box with 48x and 51x. Place 48x in the upper left square, and place 51x in the lower right square.

3	48x	51
3x	48x²	51x
	16x	17

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 48x and 51. The greatest common factor of 48x and 51 is 3. Therefore, we write 3 to the left of the top row. You can see it here in the color green:

3	48x	51
3x	48x²	51x
	16x	17

Next, let’s look at the bottom row. We have the terms 48x² and 51x. The greatest common factor of 48x² and 51x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

3	48x	51
3x	48x²	51x
	16x	17

To find the values below the table, we first divide 48x² by 3x (labeled in blue). This gives us 16x.

48x² ÷ 3x = 16x

You can see this value colored in orange below:

3	48x	51
3x	48x²	51x
	16x	17

Next, we divide 51x by 3x (labeled in blue). This gives us 17.

51x ÷ 3x = 17

You can see this value colored in purple below:

3	48x	51
3x	48x²	51x
	16x	17

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 -48x² - 99x - 51. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x + 3)(16x + 17)

In our original quadratic equation, -48x² - 99x - 51, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x + 3)(16x + 17)

That’s it! Now you know how to factor the equation -48x² - 99x - 51.

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