Factor -3x² + 83x

Factor -3x² + 83x - 54

Here we will show you how to factor the quadratic function -3x² + 83x - 54 using the box method. In other words, we will show you how to factor negative 3x squared plus 83x minus 54 (-3x^2 + 83x - 54) 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 3x² - 83x + 54. Now we can label the different parts of our equation, like this:

a = 3
b = -83
c = 54

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

-27	-81x	54
x	3x²	-2x
	3x	-2

We put 3x² (a) in the bottom left square and 54 (c) in the top right square, like this:

-27	-81x	54
x	3x²	-2x
	3x	-2

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 3 times 54 (a × c), and add together to equal -83 (b).

More specifically, 3 times 54 is 162. Therefore, we need to find the two numbers that multiply to equal 162, and add to equal -83.

? × ? = 162
? + ? = -83

After looking at this problem, we can see that the two numbers that multiply together to equal 162, and add together to equal -83, are -81 and -2, as illustrated here:

-81 × -2 = 162
-81 + -2 = -83

Now, we can fill in the last two squares in our box with -81x and -2x. Place -81x in the upper left square, and place -2x in the lower right square.

-27	-81x	54
x	3x²	-2x
	3x	-2

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 -81x and 54. The greatest common factor of -81x and 54 is -27. Therefore, we write -27 to the left of the top row. You can see it here in the color green:

-27	-81x	54
x	3x²	-2x
	3x	-2

Next, let’s look at the bottom row. We have the terms 3x² and -2x. The greatest common factor of 3x² and -2x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

-27	-81x	54
x	3x²	-2x
	3x	-2

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

3x² ÷ x = 3x

You can see this value colored in orange below:

-27	-81x	54
x	3x²	-2x
	3x	-2

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

-2x ÷ x = -2

You can see this value colored in purple below:

-27	-81x	54
x	3x²	-2x
	3x	-2

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 -3x² + 83x - 54. 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 - 27)(3x - 2)

In our original quadratic equation, -3x² + 83x - 54, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(x - 27)(3x - 2)

That’s it! Now you know how to factor the equation -3x² + 83x - 54.

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