Factor 61x² + 100x + 39

Here we will show you how to factor the quadratic function 61x² + 100x + 39 using the box method. In other words, we will show you how to factor 61x squared plus 100x plus 39 (61x^2 + 100x + 39) 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 61x² + 100x + 39, like this:

a = 61
b = 100
c = 39

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

39	39x	39
61x	61x²	61x
	x	1

We put 61x² (a) in the bottom left square and 39 (c) in the top right square, like this:

39	39x	39
61x	61x²	61x
	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 61 times 39 (a × c), and add together to equal 100 (b).

More specifically, 61 times 39 is 2379. Therefore, we need to find the two numbers that multiply to equal 2379, and add to equal 100.

? × ? = 2379
? + ? = 100

After looking at this problem, we can see that the two numbers that multiply together to equal 2379, and add together to equal 100, are 39 and 61, as illustrated here:

39 × 61 = 2379
39 + 61 = 100

Now, we can fill in the last two squares in our box with 39x and 61x. Place 39x in the upper left square, and place 61x in the lower right square.

39	39x	39
61x	61x²	61x
	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 39x and 39. The greatest common factor of 39x and 39 is 39. Therefore, we write 39 to the left of the top row. You can see it here in the color green:

39	39x	39
61x	61x²	61x
	x	1

Next, let’s look at the bottom row. We have the terms 61x² and 61x. The greatest common factor of 61x² and 61x is 61x. Therefore, we write 61x to the left of the bottom row. You can see it here in the color blue:

39	39x	39
61x	61x²	61x
	x	1

To find the values below the table, we first divide 61x² by 61x (labeled in blue). This gives us x.

61x² ÷ 61x = x

You can see this value colored in orange below:

39	39x	39
61x	61x²	61x
	x	1

Next, we divide 61x by 61x (labeled in blue). This gives us 1.

61x ÷ 61x = 1

You can see this value colored in purple below:

39	39x	39
61x	61x²	61x
	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 61x² + 100x + 39. 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:

(61x + 39)(x + 1)

That’s it! Now you know how to factor the equation 61x² + 100x + 39.

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