How to solve the cubic equation x3 − 3x2 − 9x − 7 = 0 using Cardano's formula?

To solve the cubic equation $x^3 - 3x^2 - 9x - 7 = 0$, we will first transform it into a depressed cubic equation and then use Cardano's formula.

Step 1: Transform the cubic equation into a depressed cubic.
A general cubic equation $ax^3 + bx^2 + cx + d = 0$ can be transformed into a depressed cubic $y^3 + py + q = 0$ by the substitution $x = y - \frac{b}{3a}$.
In our equation, $a=1$, $b=-3$, $c=-9$, $d=-7$.
So, we substitute $x = y - \frac{-3}{3(1)} = y + 1$.