Problem 5 - Entrance Test

Method 1: Expand directly. (x + 1/x)^2 = x^2 + 2(x)(1/x) + (1/x)^2 = x^2 + 2 + 1/x^2 (x - 1/x)^2 = x^2 - 2(x)(1/x) + (1/x)^2 = x^2 - 2 + 1/x^2 So, (x^2 + 2 + 1/x^2) - (x^2 - 2 + 1/x^2) = x^2 + 2 + 1/x^2 - x^2 + 2 - 1/x^2 = 2 + 2 = 4. Method 2: Use the difference of squares identity, A^2 - B^2 = (A+B)(A-B), where A = (x + 1/x) and B = (x - 1/x). [(x + 1/x) + (x - 1/x)][(x + 1/x) - (x - 1/x)] = [x + 1/x + x - 1/x][x + 1/x - x + 1/x] = [2x][2/x] = 4.