Factor -4x² - 28x

Factor -4x² - 28x - 48

Here we will show you how to factor the quadratic function -4x² - 28x - 48 using the box method. In other words, we will show you how to factor negative 4x squared minus 28x minus 48 (-4x^2 - 28x - 48) 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 4x² + 28x + 48. Now we can label the different parts of our equation, like this:

a = 4
b = 28
c = 48

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

12	12x	48
4x	4x²	16x
	x	4

We put 4x² (a) in the bottom left square and 48 (c) in the top right square, like this:

12	12x	48
4x	4x²	16x
	x	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 4 times 48 (a × c), and add together to equal 28 (b).

More specifically, 4 times 48 is 192. Therefore, we need to find the two numbers that multiply to equal 192, and add to equal 28.

? × ? = 192
? + ? = 28

After looking at this problem, we can see that the two numbers that multiply together to equal 192, and add together to equal 28, are 12 and 16, as illustrated here:

12 × 16 = 192
12 + 16 = 28

Now, we can fill in the last two squares in our box with 12x and 16x. Place 12x in the upper left square, and place 16x in the lower right square.

12	12x	48
4x	4x²	16x
	x	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 12x and 48. The greatest common factor of 12x and 48 is 12. Therefore, we write 12 to the left of the top row. You can see it here in the color green:

12	12x	48
4x	4x²	16x
	x	4

Next, let’s look at the bottom row. We have the terms 4x² and 16x. The greatest common factor of 4x² and 16x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

12	12x	48
4x	4x²	16x
	x	4

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

4x² ÷ 4x = x

You can see this value colored in orange below:

12	12x	48
4x	4x²	16x
	x	4

Next, we divide 16x by 4x (labeled in blue). This gives us 4.

16x ÷ 4x = 4

You can see this value colored in purple below:

12	12x	48
4x	4x²	16x
	x	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 -4x² - 28x - 48. 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 + 12)(x + 4)

In our original quadratic equation, -4x² - 28x - 48, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 12)(x + 4)

That’s it! Now you know how to factor the equation -4x² - 28x - 48.

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