Factor -92x² - 78x

Factor -92x² - 78x - 16

Here we will show you how to factor the quadratic function -92x² - 78x - 16 using the box method. In other words, we will show you how to factor negative 92x squared minus 78x minus 16 (-92x^2 - 78x - 16) 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 92x² + 78x + 16. Now we can label the different parts of our equation, like this:

a = 92
b = 78
c = 16

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

16	32x	16
46x	92x²	46x
	2x	1

We put 92x² (a) in the bottom left square and 16 (c) in the top right square, like this:

16	32x	16
46x	92x²	46x
	2x	1

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 92 times 16 (a × c), and add together to equal 78 (b).

More specifically, 92 times 16 is 1472. Therefore, we need to find the two numbers that multiply to equal 1472, and add to equal 78.

? × ? = 1472
? + ? = 78

After looking at this problem, we can see that the two numbers that multiply together to equal 1472, and add together to equal 78, are 32 and 46, as illustrated here:

32 × 46 = 1472
32 + 46 = 78

Now, we can fill in the last two squares in our box with 32x and 46x. Place 32x in the upper left square, and place 46x in the lower right square.

16	32x	16
46x	92x²	46x
	2x	1

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 32x and 16. The greatest common factor of 32x and 16 is 16. Therefore, we write 16 to the left of the top row. You can see it here in the color green:

16	32x	16
46x	92x²	46x
	2x	1

Next, let’s look at the bottom row. We have the terms 92x² and 46x. The greatest common factor of 92x² and 46x is 46x. Therefore, we write 46x to the left of the bottom row. You can see it here in the color blue:

16	32x	16
46x	92x²	46x
	2x	1

To find the values below the table, we first divide 92x² by 46x (labeled in blue). This gives us 2x.

92x² ÷ 46x = 2x

You can see this value colored in orange below:

16	32x	16
46x	92x²	46x
	2x	1

Next, we divide 46x by 46x (labeled in blue). This gives us 1.

46x ÷ 46x = 1

You can see this value colored in purple below:

16	32x	16
46x	92x²	46x
	2x	1

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 -92x² - 78x - 16. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(46x + 16)(2x + 1)

In our original quadratic equation, -92x² - 78x - 16, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(46x + 16)(2x + 1)

That’s it! Now you know how to factor the equation -92x² - 78x - 16.

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