Factor 54x² - 3x

Factor 54x² - 3x - 92

Here we will show you how to factor the quadratic function 54x² - 3x - 92 using the box method. In other words, we will show you how to factor 54x squared minus 3x minus 92 (54x^2 - 3x - 92) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 54x² - 3x - 92, like this:

a = 54
b = -3
c = -92

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

-4	-72x	-92
3x	54x²	69x
	18x	23

We put 54x² (a) in the bottom left square and -92 (c) in the top right square, like this:

-4	-72x	-92
3x	54x²	69x
	18x	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 54 times -92 (a × c), and add together to equal -3 (b).

More specifically, 54 times -92 is -4968. Therefore, we need to find the two numbers that multiply to equal -4968, and add to equal -3.

? × ? = -4968
? + ? = -3

After looking at this problem, we can see that the two numbers that multiply together to equal -4968, and add together to equal -3, are -72 and 69, as illustrated here:

-72 × 69 = -4968
-72 + 69 = -3

Now, we can fill in the last two squares in our box with -72x and 69x. Place -72x in the upper left square, and place 69x in the lower right square.

-4	-72x	-92
3x	54x²	69x
	18x	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 -72x and -92. The greatest common factor of -72x and -92 is -4. Therefore, we write -4 to the left of the top row. You can see it here in the color green:

-4	-72x	-92
3x	54x²	69x
	18x	23

Next, let’s look at the bottom row. We have the terms 54x² and 69x. The greatest common factor of 54x² and 69x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

-4	-72x	-92
3x	54x²	69x
	18x	23

To find the values below the table, we first divide 54x² by 3x (labeled in blue). This gives us 18x.

54x² ÷ 3x = 18x

You can see this value colored in orange below:

-4	-72x	-92
3x	54x²	69x
	18x	23

Next, we divide 69x by 3x (labeled in blue). This gives us 23.

69x ÷ 3x = 23

You can see this value colored in purple below:

-4	-72x	-92
3x	54x²	69x
	18x	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 54x² - 3x - 92. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(3x - 4)(18x + 23)

That’s it! Now you know how to factor the equation 54x² - 3x - 92.

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