Factor 36x² + 100x + 41

Here we will show you how to factor the quadratic function 36x² + 100x + 41 using the box method. In other words, we will show you how to factor 36x squared plus 100x plus 41 (36x^2 + 100x + 41) 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 36x² + 100x + 41, like this:

a = 36
b = 100
c = 41

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

1	18x	41
2x	36x²	82x
	18x	41

We put 36x² (a) in the bottom left square and 41 (c) in the top right square, like this:

1	18x	41
2x	36x²	82x
	18x	41

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 36 times 41 (a × c), and add together to equal 100 (b).

More specifically, 36 times 41 is 1476. Therefore, we need to find the two numbers that multiply to equal 1476, and add to equal 100.

? × ? = 1476
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal 1476, and add together to equal 100, are 18 and 82, as illustrated here:

18 × 82 = 1476
18 + 82 = 100

Now, we can fill in the last two squares in our box with 18x and 82x. Place 18x in the upper left square, and place 82x in the lower right square.

1	18x	41
2x	36x²	82x
	18x	41

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 18x and 41. The greatest common factor of 18x and 41 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	18x	41
2x	36x²	82x
	18x	41

Next, let’s look at the bottom row. We have the terms 36x² and 82x. The greatest common factor of 36x² and 82x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	18x	41
2x	36x²	82x
	18x	41

To find the values below the table, we first divide 36x² by 2x (labeled in blue). This gives us 18x.

36x² ÷ 2x = 18x

You can see this value colored in orange below:

1	18x	41
2x	36x²	82x
	18x	41

Next, we divide 82x by 2x (labeled in blue). This gives us 41.

82x ÷ 2x = 41

You can see this value colored in purple below:

1	18x	41
2x	36x²	82x
	18x	41

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 36x² + 100x + 41. 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:

(2x + 1)(18x + 41)

That’s it! Now you know how to factor the equation 36x² + 100x + 41.

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