Factor 81x² + 78x + 16

Here we will show you how to factor the quadratic function 81x² + 78x + 16 using the box method. In other words, we will show you how to factor 81x squared plus 78x plus 16 (81x^2 + 78x + 16) 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 + 78x + 16, like this:

a = 81
b = 78
c = 16

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

8	24x	16
27x	81x²	54x
	3x	2

We put 81x² (a) in the bottom left square and 16 (c) in the top right square, like this:

8	24x	16
27x	81x²	54x
	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 81 times 16 (a × c), and add together to equal 78 (b).

More specifically, 81 times 16 is 1296. Therefore, we need to find the two numbers that multiply to equal 1296, and add to equal 78.

? × ? = 1296
? + ? = 78

After looking at this problem, we can see that the two numbers that multiply together to equal 1296, and add together to equal 78, are 24 and 54, as illustrated here:

24 × 54 = 1296
24 + 54 = 78

Now, we can fill in the last two squares in our box with 24x and 54x. Place 24x in the upper left square, and place 54x in the lower right square.

8	24x	16
27x	81x²	54x
	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 24x and 16. The greatest common factor of 24x and 16 is 8. Therefore, we write 8 to the left of the top row. You can see it here in the color green:

8	24x	16
27x	81x²	54x
	3x	2

Next, let’s look at the bottom row. We have the terms 81x² and 54x. The greatest common factor of 81x² and 54x is 27x. Therefore, we write 27x to the left of the bottom row. You can see it here in the color blue:

8	24x	16
27x	81x²	54x
	3x	2

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

81x² ÷ 27x = 3x

You can see this value colored in orange below:

8	24x	16
27x	81x²	54x
	3x	2

Next, we divide 54x by 27x (labeled in blue). This gives us 2.

54x ÷ 27x = 2

You can see this value colored in purple below:

8	24x	16
27x	81x²	54x
	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 81x² + 78x + 16. 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:

(27x + 8)(3x + 2)

That’s it! Now you know how to factor the equation 81x² + 78x + 16.

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