Factor -48x² - 88x

Factor -48x² - 88x - 19

Here we will show you how to factor the quadratic function -48x² - 88x - 19 using the box method. In other words, we will show you how to factor negative 48x squared minus 88x minus 19 (-48x^2 - 88x - 19) 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 + 88x + 19. Now we can label the different parts of our equation, like this:

a = 48
b = 88
c = 19

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

1	12x	19
4x	48x²	76x
	12x	19

We put 48x² (a) in the bottom left square and 19 (c) in the top right square, like this:

1	12x	19
4x	48x²	76x
	12x	19

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 19 (a × c), and add together to equal 88 (b).

More specifically, 48 times 19 is 912. Therefore, we need to find the two numbers that multiply to equal 912, and add to equal 88.

? × ? = 912
? + ? = 88

After looking at this problem, we can see that the two numbers that multiply together to equal 912, and add together to equal 88, are 12 and 76, as illustrated here:

12 × 76 = 912
12 + 76 = 88

Now, we can fill in the last two squares in our box with 12x and 76x. Place 12x in the upper left square, and place 76x in the lower right square.

1	12x	19
4x	48x²	76x
	12x	19

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 12x and 19. The greatest common factor of 12x and 19 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	12x	19
4x	48x²	76x
	12x	19

Next, let’s look at the bottom row. We have the terms 48x² and 76x. The greatest common factor of 48x² and 76x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

1	12x	19
4x	48x²	76x
	12x	19

To find the values below the table, we first divide 48x² by 4x (labeled in blue). This gives us 12x.

48x² ÷ 4x = 12x

You can see this value colored in orange below:

1	12x	19
4x	48x²	76x
	12x	19

Next, we divide 76x by 4x (labeled in blue). This gives us 19.

76x ÷ 4x = 19

You can see this value colored in purple below:

1	12x	19
4x	48x²	76x
	12x	19

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 - 88x - 19. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x + 1)(12x + 19)

In our original quadratic equation, -48x² - 88x - 19, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 1)(12x + 19)

That’s it! Now you know how to factor the equation -48x² - 88x - 19.

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