Factor -48x² - 84x

Factor -48x² - 84x - 36

Here we will show you how to factor the quadratic function -48x² - 84x - 36 using the box method. In other words, we will show you how to factor negative 48x squared minus 84x minus 36 (-48x^2 - 84x - 36) 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 + 84x + 36. Now we can label the different parts of our equation, like this:

a = 48
b = 84
c = 36

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

36	36x	36
48x	48x²	48x
	x	1

We put 48x² (a) in the bottom left square and 36 (c) in the top right square, like this:

36	36x	36
48x	48x²	48x
	x	1

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 36 (a × c), and add together to equal 84 (b).

More specifically, 48 times 36 is 1728. Therefore, we need to find the two numbers that multiply to equal 1728, and add to equal 84.

? × ? = 1728
? + ? = 84

After looking at this problem, we can see that the two numbers that multiply together to equal 1728, and add together to equal 84, are 36 and 48, as illustrated here:

36 × 48 = 1728
36 + 48 = 84

Now, we can fill in the last two squares in our box with 36x and 48x. Place 36x in the upper left square, and place 48x in the lower right square.

36	36x	36
48x	48x²	48x
	x	1

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

36	36x	36
48x	48x²	48x
	x	1

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

36	36x	36
48x	48x²	48x
	x	1

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

48x² ÷ 48x = x

You can see this value colored in orange below:

36	36x	36
48x	48x²	48x
	x	1

Next, we divide 48x by 48x (labeled in blue). This gives us 1.

48x ÷ 48x = 1

You can see this value colored in purple below:

36	36x	36
48x	48x²	48x
	x	1

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 - 84x - 36. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(48x + 36)(x + 1)

In our original quadratic equation, -48x² - 84x - 36, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(48x + 36)(x + 1)

That’s it! Now you know how to factor the equation -48x² - 84x - 36.

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