Roots of a cubic equation

We could use the cubic discriminant , or instead, use the following trick by considering the intersection of the graphs of $y=x^3-588x$ and $\color{#4200ff}{y=5500}$ .

Differentiating the left-hand side of the equation, we obtain $3x^2-588$ . If we set this to zero, the left-hand side cubic has stationary points where $\begin{aligned}3x^2-588&=0\\\iff x^2&=\frac{588}{3}=196\\\iff x&=\pm\sqrt{196}\\&=\pm 14\end{aligned}$

From the general shape of a cubic with a positive $x^3$ coefficient, the local maximum will be where $x=-14$ (we could also consider the second derivative).

Here, $x^3-588x=(-14)^3-588\cdot (-14)=(14)^2(-14)+14(14\cdot 42)=(14)^2(42-14)=196\cdot 28=\color{#EC7300}{\boxed{5488}}$ .

Since $5488<5500$ , we can sketch the situation like this:

So there is $\color{#20A900}{\boxed{1}}$ distinct real solution (where the blue and grey lines intersect at $x>14$ ).