Given:

1. x + y = 17
2. xy = 60

We want to find x - y.

First, let's solve equation (1) for x: x = 17 - y

Now, substitute this expression for x into equation (2): (17 - y)y = 60

Expand the equation: 17y - y^2 = 60

Move all terms to one side to get a quadratic equation: y^2 - 17y + 60 = 0

Now, let's factor the quadratic equation: (y - 12)(y - 5) = 0

Setting each factor equal to zero:

1. y - 12 = 0 => y = 12
2. y - 5 = 0 => y = 5

So, we have two possible values for y, which are 12 and 5. Now, we can find x by using the equation x = 17 - y:

For y = 12: x = 17 - 12 = 5

For y = 5: x = 17 - 5 = 12

Now we have two pairs of values for x and y:

1. x = 5, y = 12
2. x = 12, y = 5

For both pairs, let's find x - y:

For x = 5, y = 12: x - y = 5 - 12 = -7

For x = 12, y = 5: x - y = 12 - 5 = 7

So, x - y can be either -7 or 7, depending on the values of x and y.