Factor 40x² + 42x + 11

Here we will show you how to factor the quadratic function 40x² + 42x + 11 using the box method. In other words, we will show you how to factor 40x squared plus 42x plus 11 (40x^2 + 42x + 11) 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 40x² + 42x + 11, like this:

a = 40
b = 42
c = 11

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

1	20x	11
2x	40x²	22x
	20x	11

We put 40x² (a) in the bottom left square and 11 (c) in the top right square, like this:

1	20x	11
2x	40x²	22x
	20x	11

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 40 times 11 (a × c), and add together to equal 42 (b).

More specifically, 40 times 11 is 440. Therefore, we need to find the two numbers that multiply to equal 440, and add to equal 42.

? × ? = 440
? + ? = 42

After looking at this problem, we can see that the two numbers that multiply together to equal 440, and add together to equal 42, are 20 and 22, as illustrated here:

20 × 22 = 440
20 + 22 = 42

Now, we can fill in the last two squares in our box with 20x and 22x. Place 20x in the upper left square, and place 22x in the lower right square.

1	20x	11
2x	40x²	22x
	20x	11

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 20x and 11. The greatest common factor of 20x and 11 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	20x	11
2x	40x²	22x
	20x	11

Next, let’s look at the bottom row. We have the terms 40x² and 22x. The greatest common factor of 40x² and 22x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	20x	11
2x	40x²	22x
	20x	11

To find the values below the table, we first divide 40x² by 2x (labeled in blue). This gives us 20x.

40x² ÷ 2x = 20x

You can see this value colored in orange below:

1	20x	11
2x	40x²	22x
	20x	11

Next, we divide 22x by 2x (labeled in blue). This gives us 11.

22x ÷ 2x = 11

You can see this value colored in purple below:

1	20x	11
2x	40x²	22x
	20x	11

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 40x² + 42x + 11. 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)(20x + 11)

That’s it! Now you know how to factor the equation 40x² + 42x + 11.

Factoring Quadratics
Go here if you want to learn how to factor another quadratic function.

Factor 40x² + 43x - 99
Here is the next quadratic function on our list that we have factored for you.