Factor -92x² + 92x

Factor -92x² + 92x - 23

Here we will show you how to factor the quadratic function -92x² + 92x - 23 using the box method. In other words, we will show you how to factor negative 92x squared plus 92x minus 23 (-92x^2 + 92x - 23) 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 - 92x + 23. Now we can label the different parts of our equation, like this:

a = 92
b = -92
c = 23

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

-23	-46x	23
46x	92x²	-46x
	2x	-1

We put 92x² (a) in the bottom left square and 23 (c) in the top right square, like this:

-23	-46x	23
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 23 (a × c), and add together to equal -92 (b).

More specifically, 92 times 23 is 2116. Therefore, we need to find the two numbers that multiply to equal 2116, and add to equal -92.

? × ? = 2116
? + ? = -92

After looking at this problem, we can see that the two numbers that multiply together to equal 2116, and add together to equal -92, are -46 and -46, as illustrated here:

-46 × -46 = 2116
-46 + -46 = -92

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

-23	-46x	23
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 -46x and 23. The greatest common factor of -46x and 23 is -23. Therefore, we write -23 to the left of the top row. You can see it here in the color green:

-23	-46x	23
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:

-23	-46x	23
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:

-23	-46x	23
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:

-23	-46x	23
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 + 92x - 23. 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 - 23)(2x - 1)

In our original quadratic equation, -92x² + 92x - 23, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(46x - 23)(2x - 1)

That’s it! Now you know how to factor the equation -92x² + 92x - 23.

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