Factor 81x² + 100x + 19

Here we will show you how to factor the quadratic function 81x² + 100x + 19 using the box method. In other words, we will show you how to factor 81x squared plus 100x plus 19 (81x^2 + 100x + 19) 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 + 100x + 19, like this:

a = 81
b = 100
c = 19

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

19	19x	19
81x	81x²	81x
	x	1

We put 81x² (a) in the bottom left square and 19 (c) in the top right square, like this:

19	19x	19
81x	81x²	81x
	x	1

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 19 (a × c), and add together to equal 100 (b).

More specifically, 81 times 19 is 1539. Therefore, we need to find the two numbers that multiply to equal 1539, and add to equal 100.

? × ? = 1539
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal 1539, and add together to equal 100, are 19 and 81, as illustrated here:

19 × 81 = 1539
19 + 81 = 100

Now, we can fill in the last two squares in our box with 19x and 81x. Place 19x in the upper left square, and place 81x in the lower right square.

19	19x	19
81x	81x²	81x
	x	1

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 19x and 19. The greatest common factor of 19x and 19 is 19. Therefore, we write 19 to the left of the top row. You can see it here in the color green:

19	19x	19
81x	81x²	81x
	x	1

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

19	19x	19
81x	81x²	81x
	x	1

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

81x² ÷ 81x = x

You can see this value colored in orange below:

19	19x	19
81x	81x²	81x
	x	1

Next, we divide 81x by 81x (labeled in blue). This gives us 1.

81x ÷ 81x = 1

You can see this value colored in purple below:

19	19x	19
81x	81x²	81x
	x	1

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 + 100x + 19. 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:

(81x + 19)(x + 1)

That’s it! Now you know how to factor the equation 81x² + 100x + 19.

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