One might say that this formula allows one to solve the quadratic with a pencil. There is an analogous formula for the general quartic equation, ax+ bx + cx + dx+ e = 0. If you just have to know the formula or one way to derive it and don’t care about anything else, you may just skip to the last section of this paper. However, there is a mathematical notion of a pencil, which is rather cool and...

In this paper, we establish the general solution of the functional equation f(nx+ y) + f(nx− y) = nf(x+ y) + nf(x− y) + 2(f(nx)− nf(x))− 2(n − 1)f(y) for fixed integers n with n 6= 0,±1 and investigate the generalized Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces.

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we deter...