Chapter 1, Equations and Graphs - Section 1.4 - Solving Quadratic Equations - 1.4 Exercises - Page 123: 52

$\frac{-1\pm\sqrt{1+8S}}{2}$

First we cross-multiply: $S= \displaystyle \frac{n(n+1)}{2}$ $2S=n^{2}+n$ $n^{2}+n-2S=0$ Now we use the quadratic formula: $n=\displaystyle \frac{-1\pm\sqrt{(1)^{2}-4(1)(-2S)}}{2(1)}=\frac{-1\pm\sqrt{1+8S}}{2}$