Factor -92x² + 19x + 1

Here we will show you how to factor the quadratic function -92x² + 19x + 1 using the box method. In other words, we will show you how to factor negative 92x squared plus 19x plus 1 (-92x^2 + 19x + 1) 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 - 19x - 1. Now we can label the different parts of our equation, like this:

a = 92
b = -19
c = -1

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

-1	-23x	-1
4x	92x²	4x
	23x	1

We put 92x² (a) in the bottom left square and -1 (c) in the top right square, like this:

-1	-23x	-1
4x	92x²	4x
	23x	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 -1 (a × c), and add together to equal -19 (b).

More specifically, 92 times -1 is -92. Therefore, we need to find the two numbers that multiply to equal -92, and add to equal -19.

? × ? = -92
? + ? = -19

After looking at this problem, we can see that the two numbers that multiply together to equal -92, and add together to equal -19, are -23 and 4, as illustrated here:

-23 × 4 = -92
-23 + 4 = -19

Now, we can fill in the last two squares in our box with -23x and 4x. Place -23x in the upper left square, and place 4x in the lower right square.

-1	-23x	-1
4x	92x²	4x
	23x	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 -23x and -1. The greatest common factor of -23x and -1 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-23x	-1
4x	92x²	4x
	23x	1

Next, let’s look at the bottom row. We have the terms 92x² and 4x. The greatest common factor of 92x² and 4x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

-1	-23x	-1
4x	92x²	4x
	23x	1

To find the values below the table, we first divide 92x² by 4x (labeled in blue). This gives us 23x.

92x² ÷ 4x = 23x

You can see this value colored in orange below:

-1	-23x	-1
4x	92x²	4x
	23x	1

Next, we divide 4x by 4x (labeled in blue). This gives us 1.

4x ÷ 4x = 1

You can see this value colored in purple below:

-1	-23x	-1
4x	92x²	4x
	23x	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 + 19x + 1. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x - 1)(23x + 1)

In our original quadratic equation, -92x² + 19x + 1, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x - 1)(23x + 1)

That’s it! Now you know how to factor the equation -92x² + 19x + 1.

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