Factor 17x² + 57x + 40

Here we will show you how to factor the quadratic function 17x² + 57x + 40 using the box method. In other words, we will show you how to factor 17x squared plus 57x plus 40 (17x^2 + 57x + 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 17x² + 57x + 40, like this:

a = 17
b = 57
c = 40

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

1	17x	40
x	17x²	40x
	17x	40

We put 17x² (a) in the bottom left square and 40 (c) in the top right square, like this:

1	17x	40
x	17x²	40x
	17x	40

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 17 times 40 (a × c), and add together to equal 57 (b).

More specifically, 17 times 40 is 680. Therefore, we need to find the two numbers that multiply to equal 680, and add to equal 57.

? × ? = 680
? + ? = 57

After looking at this problem, we can see that the two numbers that multiply together to equal 680, and add together to equal 57, are 17 and 40, as illustrated here:

17 × 40 = 680
17 + 40 = 57

Now, we can fill in the last two squares in our box with 17x and 40x. Place 17x in the upper left square, and place 40x in the lower right square.

1	17x	40
x	17x²	40x
	17x	40

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 17x and 40. The greatest common factor of 17x and 40 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	17x	40
x	17x²	40x
	17x	40

Next, let’s look at the bottom row. We have the terms 17x² and 40x. The greatest common factor of 17x² and 40x is x. Therefore, we write x to the left of the bottom row. You can see it here in the color blue:

1	17x	40
x	17x²	40x
	17x	40

To find the values below the table, we first divide 17x² by x (labeled in blue). This gives us 17x.

17x² ÷ x = 17x

You can see this value colored in orange below:

1	17x	40
x	17x²	40x
	17x	40

Next, we divide 40x by x (labeled in blue). This gives us 40.

40x ÷ x = 40

You can see this value colored in purple below:

1	17x	40
x	17x²	40x
	17x	40

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 17x² + 57x + 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:

(x + 1)(17x + 40)

That’s it! Now you know how to factor the equation 17x² + 57x + 40.

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