Factor 81x² + 99x + 28

Here we will show you how to factor the quadratic function 81x² + 99x + 28 using the box method. In other words, we will show you how to factor 81x squared plus 99x plus 28 (81x^2 + 99x + 28) 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 + 99x + 28, like this:

a = 81
b = 99
c = 28

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

4	36x	28
9x	81x²	63x
	9x	7

We put 81x² (a) in the bottom left square and 28 (c) in the top right square, like this:

4	36x	28
9x	81x²	63x
	9x	7

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 28 (a × c), and add together to equal 99 (b).

More specifically, 81 times 28 is 2268. Therefore, we need to find the two numbers that multiply to equal 2268, and add to equal 99.

? × ? = 2268
? + ? = 99

After looking at this problem, we can see that the two numbers that multiply together to equal 2268, and add together to equal 99, are 36 and 63, as illustrated here:

36 × 63 = 2268
36 + 63 = 99

Now, we can fill in the last two squares in our box with 36x and 63x. Place 36x in the upper left square, and place 63x in the lower right square.

4	36x	28
9x	81x²	63x
	9x	7

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 36x and 28. The greatest common factor of 36x and 28 is 4. Therefore, we write 4 to the left of the top row. You can see it here in the color green:

4	36x	28
9x	81x²	63x
	9x	7

Next, let’s look at the bottom row. We have the terms 81x² and 63x. The greatest common factor of 81x² and 63x is 9x. Therefore, we write 9x to the left of the bottom row. You can see it here in the color blue:

4	36x	28
9x	81x²	63x
	9x	7

To find the values below the table, we first divide 81x² by 9x (labeled in blue). This gives us 9x.

81x² ÷ 9x = 9x

You can see this value colored in orange below:

4	36x	28
9x	81x²	63x
	9x	7

Next, we divide 63x by 9x (labeled in blue). This gives us 7.

63x ÷ 9x = 7

You can see this value colored in purple below:

4	36x	28
9x	81x²	63x
	9x	7

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 + 99x + 28. 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:

(9x + 4)(9x + 7)

That’s it! Now you know how to factor the equation 81x² + 99x + 28.

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