Factor 60x² + 64x + 17

Here we will show you how to factor the quadratic function 60x² + 64x + 17 using the box method. In other words, we will show you how to factor 60x squared plus 64x plus 17 (60x^2 + 64x + 17) 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 60x² + 64x + 17, like this:

a = 60
b = 64
c = 17

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

1	30x	17
2x	60x²	34x
	30x	17

We put 60x² (a) in the bottom left square and 17 (c) in the top right square, like this:

1	30x	17
2x	60x²	34x
	30x	17

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 60 times 17 (a × c), and add together to equal 64 (b).

More specifically, 60 times 17 is 1020. Therefore, we need to find the two numbers that multiply to equal 1020, and add to equal 64.

? × ? = 1020
? + ? = 64

After looking at this problem, we can see that the two numbers that multiply together to equal 1020, and add together to equal 64, are 30 and 34, as illustrated here:

30 × 34 = 1020
30 + 34 = 64

Now, we can fill in the last two squares in our box with 30x and 34x. Place 30x in the upper left square, and place 34x in the lower right square.

1	30x	17
2x	60x²	34x
	30x	17

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 30x and 17. The greatest common factor of 30x and 17 is 1. Therefore, we write 1 to the left of the top row. You can see it here in the color green:

1	30x	17
2x	60x²	34x
	30x	17

Next, let’s look at the bottom row. We have the terms 60x² and 34x. The greatest common factor of 60x² and 34x is 2x. Therefore, we write 2x to the left of the bottom row. You can see it here in the color blue:

1	30x	17
2x	60x²	34x
	30x	17

To find the values below the table, we first divide 60x² by 2x (labeled in blue). This gives us 30x.

60x² ÷ 2x = 30x

You can see this value colored in orange below:

1	30x	17
2x	60x²	34x
	30x	17

Next, we divide 34x by 2x (labeled in blue). This gives us 17.

34x ÷ 2x = 17

You can see this value colored in purple below:

1	30x	17
2x	60x²	34x
	30x	17

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 60x² + 64x + 17. 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)(30x + 17)

That’s it! Now you know how to factor the equation 60x² + 64x + 17.

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