jmc

Question

The equation x^3 - 4x^2 + 5x - 1910 = 0 has real roots r, s, and t. Find the length of the long diagonal of a box with sides of length r, s, and t.

Accepted Answer

The length of the long diagonal is r^2 + s^2 + t^2. By Vieta's formulas, r + s + t = 4 and rs + rt + st = 5. Squaring the equation r + s + t = 4, we get r^2 + s^2 + t^2 + 2(rs + rt + st) = 16.Then r^2 + s^2 + t^2 = 16 - 2(rs + rt + st) = 6, so r^2 + s^2 + t^2 = 6.