Factor -48x² - 71x

Factor -48x² - 71x - 23

Here we will show you how to factor the quadratic function -48x² - 71x - 23 using the box method. In other words, we will show you how to factor negative 48x squared minus 71x minus 23 (-48x^2 - 71x - 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 48x² + 71x + 23. Now we can label the different parts of our equation, like this:

a = 48
b = 71
c = 23

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

23	23x	23
48x	48x²	48x
	x	1

We put 48x² (a) in the bottom left square and 23 (c) in the top right square, like this:

23	23x	23
48x	48x²	48x
	x	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 48 times 23 (a × c), and add together to equal 71 (b).

More specifically, 48 times 23 is 1104. Therefore, we need to find the two numbers that multiply to equal 1104, and add to equal 71.

? × ? = 1104
? + ? = 71

After looking at this problem, we can see that the two numbers that multiply together to equal 1104, and add together to equal 71, are 23 and 48, as illustrated here:

23 × 48 = 1104
23 + 48 = 71

Now, we can fill in the last two squares in our box with 23x and 48x. Place 23x in the upper left square, and place 48x in the lower right square.

23	23x	23
48x	48x²	48x
	x	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 23. The greatest common factor of 23x 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	23x	23
48x	48x²	48x
	x	1

Next, let’s look at the bottom row. We have the terms 48x² and 48x. The greatest common factor of 48x² and 48x is 48x. Therefore, we write 48x to the left of the bottom row. You can see it here in the color blue:

23	23x	23
48x	48x²	48x
	x	1

To find the values below the table, we first divide 48x² by 48x (labeled in blue). This gives us x.

48x² ÷ 48x = x

You can see this value colored in orange below:

23	23x	23
48x	48x²	48x
	x	1

Next, we divide 48x by 48x (labeled in blue). This gives us 1.

48x ÷ 48x = 1

You can see this value colored in purple below:

23	23x	23
48x	48x²	48x
	x	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 -48x² - 71x - 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:

(48x + 23)(x + 1)

In our original quadratic equation, -48x² - 71x - 23, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(48x + 23)(x + 1)

That’s it! Now you know how to factor the equation -48x² - 71x - 23.

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