Factor 39x² + 86x + 40

Here we will show you how to factor the quadratic function 39x² + 86x + 40 using the box method. In other words, we will show you how to factor 39x squared plus 86x plus 40 (39x^2 + 86x + 40) 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 39x² + 86x + 40, like this:

a = 39
b = 86
c = 40

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

2	26x	40
3x	39x²	60x
	13x	20

We put 39x² (a) in the bottom left square and 40 (c) in the top right square, like this:

2	26x	40
3x	39x²	60x
	13x	20

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 39 times 40 (a × c), and add together to equal 86 (b).

More specifically, 39 times 40 is 1560. Therefore, we need to find the two numbers that multiply to equal 1560, and add to equal 86.

? × ? = 1560
? + ? = 86

After looking at this problem, we can see that the two numbers that multiply together to equal 1560, and add together to equal 86, are 26 and 60, as illustrated here:

26 × 60 = 1560
26 + 60 = 86

Now, we can fill in the last two squares in our box with 26x and 60x. Place 26x in the upper left square, and place 60x in the lower right square.

2	26x	40
3x	39x²	60x
	13x	20

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 26x and 40. The greatest common factor of 26x and 40 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	26x	40
3x	39x²	60x
	13x	20

Next, let’s look at the bottom row. We have the terms 39x² and 60x. The greatest common factor of 39x² and 60x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

2	26x	40
3x	39x²	60x
	13x	20

To find the values below the table, we first divide 39x² by 3x (labeled in blue). This gives us 13x.

39x² ÷ 3x = 13x

You can see this value colored in orange below:

2	26x	40
3x	39x²	60x
	13x	20

Next, we divide 60x by 3x (labeled in blue). This gives us 20.

60x ÷ 3x = 20

You can see this value colored in purple below:

2	26x	40
3x	39x²	60x
	13x	20

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 39x² + 86x + 40. 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 + 2)(13x + 20)

That’s it! Now you know how to factor the equation 39x² + 86x + 40.

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