Factor -12x² - 52x

Factor -12x² - 52x - 23

Here we will show you how to factor the quadratic function -12x² - 52x - 23 using the box method. In other words, we will show you how to factor negative 12x squared minus 52x minus 23 (-12x^2 - 52x - 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 12x² + 52x + 23. Now we can label the different parts of our equation, like this:

a = 12
b = 52
c = 23

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

1	6x	23
2x	12x²	46x
	6x	23

We put 12x² (a) in the bottom left square and 23 (c) in the top right square, like this:

1	6x	23
2x	12x²	46x
	6x	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 12 times 23 (a × c), and add together to equal 52 (b).

More specifically, 12 times 23 is 276. Therefore, we need to find the two numbers that multiply to equal 276, and add to equal 52.

? × ? = 276
? + ? = 52

After looking at this problem, we can see that the two numbers that multiply together to equal 276, and add together to equal 52, are 6 and 46, as illustrated here:

6 × 46 = 276
6 + 46 = 52

Now, we can fill in the last two squares in our box with 6x and 46x. Place 6x in the upper left square, and place 46x in the lower right square.

1	6x	23
2x	12x²	46x
	6x	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 6x and 23. The greatest common factor of 6x 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	6x	23
2x	12x²	46x
	6x	23

Next, let’s look at the bottom row. We have the terms 12x² and 46x. The greatest common factor of 12x² and 46x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	6x	23
2x	12x²	46x
	6x	23

To find the values below the table, we first divide 12x² by 2x (labeled in blue). This gives us 6x.

12x² ÷ 2x = 6x

You can see this value colored in orange below:

1	6x	23
2x	12x²	46x
	6x	23

Next, we divide 46x by 2x (labeled in blue). This gives us 23.

46x ÷ 2x = 23

You can see this value colored in purple below:

1	6x	23
2x	12x²	46x
	6x	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 -12x² - 52x - 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:

(2x + 1)(6x + 23)

In our original quadratic equation, -12x² - 52x - 23, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(2x + 1)(6x + 23)

That’s it! Now you know how to factor the equation -12x² - 52x - 23.

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