Factor -7x² - 38x

Factor -7x² - 38x - 48

Here we will show you how to factor the quadratic function -7x² - 38x - 48 using the box method. In other words, we will show you how to factor negative 7x squared minus 38x minus 48 (-7x^2 - 38x - 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 7x² + 38x + 48. Now we can label the different parts of our equation, like this:

a = 7
b = 38
c = 48

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

2	14x	48
x	7x²	24x
	7x	24

We put 7x² (a) in the bottom left square and 48 (c) in the top right square, like this:

2	14x	48
x	7x²	24x
	7x	24

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 7 times 48 (a × c), and add together to equal 38 (b).

More specifically, 7 times 48 is 336. Therefore, we need to find the two numbers that multiply to equal 336, and add to equal 38.

? × ? = 336
? + ? = 38

After looking at this problem, we can see that the two numbers that multiply together to equal 336, and add together to equal 38, are 14 and 24, as illustrated here:

14 × 24 = 336
14 + 24 = 38

Now, we can fill in the last two squares in our box with 14x and 24x. Place 14x in the upper left square, and place 24x in the lower right square.

2	14x	48
x	7x²	24x
	7x	24

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 14x and 48. The greatest common factor of 14x and 48 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	14x	48
x	7x²	24x
	7x	24

Next, let’s look at the bottom row. We have the terms 7x² and 24x. The greatest common factor of 7x² and 24x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

2	14x	48
x	7x²	24x
	7x	24

To find the values below the table, we first divide 7x² by x (labeled in blue). This gives us 7x.

7x² ÷ x = 7x

You can see this value colored in orange below:

2	14x	48
x	7x²	24x
	7x	24

Next, we divide 24x by x (labeled in blue). This gives us 24.

24x ÷ x = 24

You can see this value colored in purple below:

2	14x	48
x	7x²	24x
	7x	24

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 -7x² - 38x - 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:

(x + 2)(7x + 24)

In our original quadratic equation, -7x² - 38x - 48, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 2)(7x + 24)

That’s it! Now you know how to factor the equation -7x² - 38x - 48.

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