Solve the following equation n2 652 65n2 65 64 0 Select

Solve the following equation. (n^2 + 65)^2 - 65(n^2 + 65) + 64 = 0 Select the correct choice below and, if necessary, fill in the choice. The solution(s) is/are

Solution

We have (n²+65)²-65(n²+65)+64 = 0. Let n²+65 = x. Then the equation changes to x² -65x +64 = 0 or,x² -64x -x+64 = 0 or, x(x-64) -1(x-64) = 0 or, (x-64)(x-1) = 0. Therefore x = 64 or x = 1. If x = 64, then n²+65 = 64 so that n² = 64-65 = -1. Then n = ± i. Also, if x = 1, then n²+65 = 1 or n² = 1-65 = -64 so that n = ± 8i. Thus, the solutions are n = i, -i, 8i and -8i.