Factor 81x² + 75x + 16

Here we will show you how to factor the quadratic function 81x² + 75x + 16 using the box method. In other words, we will show you how to factor 81x squared plus 75x plus 16 (81x^2 + 75x + 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 + 75x + 16, like this:

a = 81
b = 75
c = 16

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

1	27x	16
3x	81x²	48x
	27x	16

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

1	27x	16
3x	81x²	48x
	27x	16

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 75 (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 75.

? × ? = 1296
? + ? = 75

After looking at this problem, we can see that the two numbers that multiply together to equal 1296, and add together to equal 75, are 27 and 48, as illustrated here:

27 × 48 = 1296
27 + 48 = 75

Now, we can fill in the last two squares in our box with 27x and 48x. Place 27x in the upper left square, and place 48x in the lower right square.

1	27x	16
3x	81x²	48x
	27x	16

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 16. The greatest common factor of 27x and 16 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	27x	16
3x	81x²	48x
	27x	16

Next, let’s look at the bottom row. We have the terms 81x² and 48x. The greatest common factor of 81x² and 48x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

1	27x	16
3x	81x²	48x
	27x	16

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	16
3x	81x²	48x
	27x	16

Next, we divide 48x by 3x (labeled in blue). This gives us 16.

48x ÷ 3x = 16

You can see this value colored in purple below:

1	27x	16
3x	81x²	48x
	27x	16

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 + 75x + 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:

(3x + 1)(27x + 16)

That’s it! Now you know how to factor the equation 81x² + 75x + 16.

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