Factor -21x² - 61x + 92

Here we will show you how to factor the quadratic function -21x² - 61x + 92 using the box method. In other words, we will show you how to factor negative 21x squared minus 61x plus 92 (-21x^2 - 61x + 92) 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 21x² + 61x - 92. Now we can label the different parts of our equation, like this:

a = 21
b = 61
c = -92

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

-23	-23x	-92
21x	21x²	84x
	x	4

We put 21x² (a) in the bottom left square and -92 (c) in the top right square, like this:

-23	-23x	-92
21x	21x²	84x
	x	4

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 21 times -92 (a × c), and add together to equal 61 (b).

More specifically, 21 times -92 is -1932. Therefore, we need to find the two numbers that multiply to equal -1932, and add to equal 61.

? × ? = -1932
? + ? = 61

After looking at this problem, we can see that the two numbers that multiply together to equal -1932, and add together to equal 61, are -23 and 84, as illustrated here:

-23 × 84 = -1932
-23 + 84 = 61

Now, we can fill in the last two squares in our box with -23x and 84x. Place -23x in the upper left square, and place 84x in the lower right square.

-23	-23x	-92
21x	21x²	84x
	x	4

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

-23	-23x	-92
21x	21x²	84x
	x	4

Next, let’s look at the bottom row. We have the terms 21x² and 84x. The greatest common factor of 21x² and 84x is 21x. Therefore, we write 21x to the left of the bottom row. You can see it here in the color blue:

-23	-23x	-92
21x	21x²	84x
	x	4

To find the values below the table, we first divide 21x² by 21x (labeled in blue). This gives us x.

21x² ÷ 21x = x

You can see this value colored in orange below:

-23	-23x	-92
21x	21x²	84x
	x	4

Next, we divide 84x by 21x (labeled in blue). This gives us 4.

84x ÷ 21x = 4

You can see this value colored in purple below:

-23	-23x	-92
21x	21x²	84x
	x	4

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 -21x² - 61x + 92. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(21x - 23)(x + 4)

In our original quadratic equation, -21x² - 61x + 92, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(21x - 23)(x + 4)

That’s it! Now you know how to factor the equation -21x² - 61x + 92.

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