If x^2 - y^2 = 16 and xy = 15, find out x + y ?

Strategy: use symmetric sums. Let s = x + y and p = xy = 15.

Step 1: Express x^2 - y^2 in terms of s.

Since x^2 - y^2 = (x + y)(x - y) = 16, we have x - y = 16/s.

Step 2: Use (x - y)^2 = s^2 - 4p.

So (16/s)^2 = s^2 - 4·15 = s^2 - 60. Therefore 256/s^2 = s^2 - 60.

Step 3: Multiply through by s^2 and set t = s^2 to solve:

• s^4 - 60s^2 - 256 = 0, so t^2 - 60t - 256 = 0 where t = s^2.

• Discriminant = 60^2 + 4·256 = 3600 + 1024 = 4624 = 68^2.

• t = (60 ± 68)/2 gives t = 64 or t = -4. Discard t = -4 since t = s^2 ≥ 0, so s^2 = 64.

Step 4: Take square roots to get s = x + y = ±8.

Conclusion: The possible sums are 8 and -8. For the integer pair x = 5, y = 3 we get x + y = 8 (and x = -5, y = -3 gives -8). Among the provided choices, 8 is the correct answer.