Factor 81x² + 93x + 22

Here we will show you how to factor the quadratic function 81x² + 93x + 22 using the box method. In other words, we will show you how to factor 81x squared plus 93x plus 22 (81x^2 + 93x + 22) 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 81x² + 93x + 22, like this:

a = 81
b = 93
c = 22

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

1	27x	22
3x	81x²	66x
	27x	22

We put 81x² (a) in the bottom left square and 22 (c) in the top right square, like this:

1	27x	22
3x	81x²	66x
	27x	22

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 81 times 22 (a × c), and add together to equal 93 (b).

More specifically, 81 times 22 is 1782. Therefore, we need to find the two numbers that multiply to equal 1782, and add to equal 93.

? × ? = 1782
? + ? = 93

After looking at this problem, we can see that the two numbers that multiply together to equal 1782, and add together to equal 93, are 27 and 66, as illustrated here:

27 × 66 = 1782
27 + 66 = 93

Now, we can fill in the last two squares in our box with 27x and 66x. Place 27x in the upper left square, and place 66x in the lower right square.

1	27x	22
3x	81x²	66x
	27x	22

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 27x and 22. The greatest common factor of 27x and 22 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	27x	22
3x	81x²	66x
	27x	22

Next, let’s look at the bottom row. We have the terms 81x² and 66x. The greatest common factor of 81x² and 66x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

1	27x	22
3x	81x²	66x
	27x	22

To find the values below the table, we first divide 81x² by 3x (labeled in blue). This gives us 27x.

81x² ÷ 3x = 27x

You can see this value colored in orange below:

1	27x	22
3x	81x²	66x
	27x	22

Next, we divide 66x by 3x (labeled in blue). This gives us 22.

66x ÷ 3x = 22

You can see this value colored in purple below:

1	27x	22
3x	81x²	66x
	27x	22

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 81x² + 93x + 22. 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 + 1)(27x + 22)

That’s it! Now you know how to factor the equation 81x² + 93x + 22.

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