Factor 40x² + 41x + 10

Here we will show you how to factor the quadratic function 40x² + 41x + 10 using the box method. In other words, we will show you how to factor 40x squared plus 41x plus 10 (40x^2 + 41x + 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 + 41x + 10, like this:

a = 40
b = 41
c = 10

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

2	16x	10
5x	40x²	25x
	8x	5

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

2	16x	10
5x	40x²	25x
	8x	5

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 41 (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 41.

? × ? = 400
? + ? = 41

After looking at this problem, we can see that the two numbers that multiply together to equal 400, and add together to equal 41, are 16 and 25, as illustrated here:

16 × 25 = 400
16 + 25 = 41

Now, we can fill in the last two squares in our box with 16x and 25x. Place 16x in the upper left square, and place 25x in the lower right square.

2	16x	10
5x	40x²	25x
	8x	5

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 16x and 10. The greatest common factor of 16x and 10 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	16x	10
5x	40x²	25x
	8x	5

Next, let’s look at the bottom row. We have the terms 40x² and 25x. The greatest common factor of 40x² and 25x is 5x. Therefore, we write 5x to the left of the bottom row. You can see it here in the color blue:

2	16x	10
5x	40x²	25x
	8x	5

To find the values below the table, we first divide 40x² by 5x (labeled in blue). This gives us 8x.

40x² ÷ 5x = 8x

You can see this value colored in orange below:

2	16x	10
5x	40x²	25x
	8x	5

Next, we divide 25x by 5x (labeled in blue). This gives us 5.

25x ÷ 5x = 5

You can see this value colored in purple below:

2	16x	10
5x	40x²	25x
	8x	5

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 + 41x + 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:

(5x + 2)(8x + 5)

That’s it! Now you know how to factor the equation 40x² + 41x + 10.

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