  Omar Khayyam's studies on cubic equations inspired the 12th century Persian
mathematician Sharaf al-Din Tusi to investigate the number of positive roots.
According to the noted mathematical historian Rashed, Tusi analyzed the problem
for five different types of equations. In fact all cubic equations are
reducible to a form {\it Tusi form} $x^2-x^3=c$. Tusi determined that the
maximum of $x^2-x^3$ on $(0,1)$ occurs at $\frac{2}{3}$ and concluded when
$c=\frac{4}{27} \delta$, $\delta \in (0,1)$, there are roots in $(0,
\frac{2}{3})$ and $(\frac{2}{3},1)$, ignoring the root in $(-\frac{1}{3},0)$.
Given a {\it reduced form} $x^3+px+q=0$, when $p <0$, we show it is reducible
to a Tusi form with $\delta = \frac{1}{2} + {3\sqrt{3} q}/{4\sqrt{-p^3}}$. It
follows there are three real roots if and only if $\Delta
=-(\frac{q^2}{4}+\frac{p^3}{27})$ is positive. This gives an explicit
connection between $\delta$ in Tusi form and $\Delta$ in Cardano's formula.
Thus when $\delta \in (0,1)$, rather than using Cardano's formula in complex
numbers one can approximate the roots iteratively. On the other hand, for a
reduced form with $p >0$ we give a novel proof of Cardono's formula. While
Rashed attributes Tusi's computation of the maximum to the use of derivatives,
according to Hogendijk, Tusi was probably influenced by Euclid. Here we show
the maximizer in Tusi form is computable via elementary algebraic
manipulations. Indeed for a {\it quadratic Tusi form}, $x-x^2=\delta/4$, Tusi's
approach results in a simple derivation of the quadratic formula, comparable
with the pedagogical approach of Po-Shen Loh. Moreover, we derive analogous
results for the {\it general Tusi form}. Finally, we present a novel derivation
of Khayyam's geometric solution. The results complement previous findings on
Tusi's work and reveal further facts on history, mathematics and pedagogy in
solving cubic equations.
