Factor -11x² - 12x + 23

Here we will show you how to factor the quadratic function -11x² - 12x + 23 using the box method. In other words, we will show you how to factor negative 11x squared minus 12x plus 23 (-11x^2 - 12x + 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 11x² + 12x - 23. Now we can label the different parts of our equation, like this:

a = 11
b = 12
c = -23

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

-1	-11x	-23
x	11x²	23x
	11x	23

We put 11x² (a) in the bottom left square and -23 (c) in the top right square, like this:

-1	-11x	-23
x	11x²	23x
	11x	23

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 11 times -23 (a × c), and add together to equal 12 (b).

More specifically, 11 times -23 is -253. Therefore, we need to find the two numbers that multiply to equal -253, and add to equal 12.

? × ? = -253
? + ? = 12

After looking at this problem, we can see that the two numbers that multiply together to equal -253, and add together to equal 12, are -11 and 23, as illustrated here:

-11 × 23 = -253
-11 + 23 = 12

Now, we can fill in the last two squares in our box with -11x and 23x. Place -11x in the upper left square, and place 23x in the lower right square.

-1	-11x	-23
x	11x²	23x
	11x	23

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 -11x and -23. The greatest common factor of -11x and -23 is -1. Therefore, we write -1 to the left of the top row. You can see it here in the color green:

-1	-11x	-23
x	11x²	23x
	11x	23

Next, let’s look at the bottom row. We have the terms 11x² and 23x. The greatest common factor of 11x² and 23x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-1	-11x	-23
x	11x²	23x
	11x	23

To find the values below the table, we first divide 11x² by x (labeled in blue). This gives us 11x.

11x² ÷ x = 11x

You can see this value colored in orange below:

-1	-11x	-23
x	11x²	23x
	11x	23

Next, we divide 23x by x (labeled in blue). This gives us 23.

23x ÷ x = 23

You can see this value colored in purple below:

-1	-11x	-23
x	11x²	23x
	11x	23

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 -11x² - 12x + 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:

(x - 1)(11x + 23)

In our original quadratic equation, -11x² - 12x + 23, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 1)(11x + 23)

That’s it! Now you know how to factor the equation -11x² - 12x + 23.

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