Factor -16x² - 47x

Factor -16x² - 47x - 31

Here we will show you how to factor the quadratic function -16x² - 47x - 31 using the box method. In other words, we will show you how to factor negative 16x squared minus 47x minus 31 (-16x^2 - 47x - 31) 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 + 47x + 31. Now we can label the different parts of our equation, like this:

a = 16
b = 47
c = 31

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

1	16x	31
x	16x²	31x
	16x	31

We put 16x² (a) in the bottom left square and 31 (c) in the top right square, like this:

1	16x	31
x	16x²	31x
	16x	31

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 31 (a × c), and add together to equal 47 (b).

More specifically, 16 times 31 is 496. Therefore, we need to find the two numbers that multiply to equal 496, and add to equal 47.

? × ? = 496
? + ? = 47

After looking at this problem, we can see that the two numbers that multiply together to equal 496, and add together to equal 47, are 16 and 31, as illustrated here:

16 × 31 = 496
16 + 31 = 47

Now, we can fill in the last two squares in our box with 16x and 31x. Place 16x in the upper left square, and place 31x in the lower right square.

1	16x	31
x	16x²	31x
	16x	31

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 31. The greatest common factor of 16x and 31 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	16x	31
x	16x²	31x
	16x	31

Next, let’s look at the bottom row. We have the terms 16x² and 31x. The greatest common factor of 16x² and 31x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	16x	31
x	16x²	31x
	16x	31

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

16x² ÷ x = 16x

You can see this value colored in orange below:

1	16x	31
x	16x²	31x
	16x	31

Next, we divide 31x by x (labeled in blue). This gives us 31.

31x ÷ x = 31

You can see this value colored in purple below:

1	16x	31
x	16x²	31x
	16x	31

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 - 47x - 31. 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 + 1)(16x + 31)

In our original quadratic equation, -16x² - 47x - 31, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x + 1)(16x + 31)

That’s it! Now you know how to factor the equation -16x² - 47x - 31.

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Factor -16x² - 47x - 30
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