Sum of Roots of Cubic Equation

Theorem
Let $P$ be the cubic equation $a x^3 + b x^2 + c x + d = 0$.

Let $\alpha, \beta, \gamma$ be the roots of $P$.

Then:
 * $\alpha + \beta + \gamma = - \dfrac b a$

Proof
From Cardano's Formula, $P$ has solutions:
 * $\alpha = S + T - \dfrac b {3 a}$
 * $\beta = - \dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \left({S - T}\right)$
 * $\gamma = - \dfrac {S + T} 2 - \dfrac b {3 a} - \dfrac {i \sqrt 3} 2 \left({S - T}\right)$

where:
 * $S = \sqrt [3] {R + \sqrt{Q^3 + R^2}}$
 * $T = \sqrt [3] {R - \sqrt{Q^3 + R^2}}$

where:
 * $Q = \dfrac {3 a c - b^2} {9 a^2}$
 * $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

Thus: