Factor -48x² - 91x + 82

Here we will show you how to factor the quadratic function -48x² - 91x + 82 using the box method. In other words, we will show you how to factor negative 48x squared minus 91x plus 82 (-48x^2 - 91x + 82) 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 + 91x - 82. Now we can label the different parts of our equation, like this:

a = 48
b = 91
c = -82

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

-2	-32x	-82
3x	48x²	123x
	16x	41

We put 48x² (a) in the bottom left square and -82 (c) in the top right square, like this:

-2	-32x	-82
3x	48x²	123x
	16x	41

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 -82 (a × c), and add together to equal 91 (b).

More specifically, 48 times -82 is -3936. Therefore, we need to find the two numbers that multiply to equal -3936, and add to equal 91.

? × ? = -3936
? + ? = 91

After looking at this problem, we can see that the two numbers that multiply together to equal -3936, and add together to equal 91, are -32 and 123, as illustrated here:

-32 × 123 = -3936
-32 + 123 = 91

Now, we can fill in the last two squares in our box with -32x and 123x. Place -32x in the upper left square, and place 123x in the lower right square.

-2	-32x	-82
3x	48x²	123x
	16x	41

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 -32x and -82. The greatest common factor of -32x and -82 is -2. Therefore, we write -2 to the left of the top row. You can see it here in the color green:

-2	-32x	-82
3x	48x²	123x
	16x	41

Next, let’s look at the bottom row. We have the terms 48x² and 123x. The greatest common factor of 48x² and 123x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-2	-32x	-82
3x	48x²	123x
	16x	41

To find the values below the table, we first divide 48x² by 3x (labeled in blue). This gives us 16x.

48x² ÷ 3x = 16x

You can see this value colored in orange below:

-2	-32x	-82
3x	48x²	123x
	16x	41

Next, we divide 123x by 3x (labeled in blue). This gives us 41.

123x ÷ 3x = 41

You can see this value colored in purple below:

-2	-32x	-82
3x	48x²	123x
	16x	41

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 - 91x + 82. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(3x - 2)(16x + 41)

In our original quadratic equation, -48x² - 91x + 82, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(3x - 2)(16x + 41)

That’s it! Now you know how to factor the equation -48x² - 91x + 82.

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Factor -48x² - 90x - 42
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