Factor 40x² + 40x + 10

Here we will show you how to factor the quadratic function 40x² + 40x + 10 using the box method. In other words, we will show you how to factor 40x squared plus 40x plus 10 (40x^2 + 40x + 10) 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 + 40x + 10, like this:

a = 40
b = 40
c = 10

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

10	20x	10
20x	40x²	20x
	2x	1

We put 40x² (a) in the bottom left square and 10 (c) in the top right square, like this:

10	20x	10
20x	40x²	20x
	2x	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 40 times 10 (a × c), and add together to equal 40 (b).

More specifically, 40 times 10 is 400. Therefore, we need to find the two numbers that multiply to equal 400, and add to equal 40.

? × ? = 400
? + ? = 40

After looking at this problem, we can see that the two numbers that multiply together to equal 400, and add together to equal 40, are 20 and 20, as illustrated here:

20 × 20 = 400
20 + 20 = 40

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

10	20x	10
20x	40x²	20x
	2x	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 20x and 10. The greatest common factor of 20x and 10 is 10. Therefore, we write 10 to the left of the top row. You can see it here in the color green:

10	20x	10
20x	40x²	20x
	2x	1

Next, let’s look at the bottom row. We have the terms 40x² and 20x. The greatest common factor of 40x² and 20x is 20x. Therefore, we write 20x to the left of the bottom row. You can see it here in the color blue:

10	20x	10
20x	40x²	20x
	2x	1

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

40x² ÷ 20x = 2x

You can see this value colored in orange below:

10	20x	10
20x	40x²	20x
	2x	1

Next, we divide 20x by 20x (labeled in blue). This gives us 1.

20x ÷ 20x = 1

You can see this value colored in purple below:

10	20x	10
20x	40x²	20x
	2x	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 40x² + 40x + 10. 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:

(20x + 10)(2x + 1)

That’s it! Now you know how to factor the equation 40x² + 40x + 10.

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