Factor -16x² - 54x

Factor -16x² - 54x - 38

Here we will show you how to factor the quadratic function -16x² - 54x - 38 using the box method. In other words, we will show you how to factor negative 16x squared minus 54x minus 38 (-16x^2 - 54x - 38) 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 16x² + 54x + 38. Now we can label the different parts of our equation, like this:

a = 16
b = 54
c = 38

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

2	16x	38
2x	16x²	38x
	8x	19

We put 16x² (a) in the bottom left square and 38 (c) in the top right square, like this:

2	16x	38
2x	16x²	38x
	8x	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 16 times 38 (a × c), and add together to equal 54 (b).

More specifically, 16 times 38 is 608. Therefore, we need to find the two numbers that multiply to equal 608, and add to equal 54.

? × ? = 608
? + ? = 54

After looking at this problem, we can see that the two numbers that multiply together to equal 608, and add together to equal 54, are 16 and 38, as illustrated here:

16 × 38 = 608
16 + 38 = 54

Now, we can fill in the last two squares in our box with 16x and 38x. Place 16x in the upper left square, and place 38x in the lower right square.

2	16x	38
2x	16x²	38x
	8x	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 16x and 38. The greatest common factor of 16x and 38 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	16x	38
2x	16x²	38x
	8x	19

Next, let’s look at the bottom row. We have the terms 16x² and 38x. The greatest common factor of 16x² and 38x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

2	16x	38
2x	16x²	38x
	8x	19

To find the values below the table, we first divide 16x² by 2x (labeled in blue). This gives us 8x.

16x² ÷ 2x = 8x

You can see this value colored in orange below:

2	16x	38
2x	16x²	38x
	8x	19

Next, we divide 38x by 2x (labeled in blue). This gives us 19.

38x ÷ 2x = 19

You can see this value colored in purple below:

2	16x	38
2x	16x²	38x
	8x	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 -16x² - 54x - 38. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(2x + 2)(8x + 19)

In our original quadratic equation, -16x² - 54x - 38, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 2)(8x + 19)

That’s it! Now you know how to factor the equation -16x² - 54x - 38.

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