Cardano's Formula Calculator

Step-by-Step Solution:

1. Input the coefficients \( a \), \( b \), \( c \), and \( d \).
2. Calculate \( p \) and \( q \):
   • \( p = \frac{-b^2}{3a^2} + \frac{c}{a} \)
   • \( q = \frac{2b^3}{27a^3} - \frac{bc}{3a^2} + \frac{d}{a} \)
3. Calculate the discriminant \( \Delta = \frac{q}{2}^2 + \frac{p}{3}^3 \).
4. Check \( \Delta \) value and determine the number of real solutions:
   • If \( \Delta > 0 \), there is one real solution and two complex solutions.
   • If \( \Delta = 0 \), there are two real solutions and one repeated real solution.
   • If \( \Delta < 0 \), there are three distinct real solutions.
5. Solve for solutions depending on the case:
   • For \( \Delta > 0 \):
     • Calculate \( u = \sqrt[3]{-\frac{q}{2} + \sqrt{\Delta}} \) and \( v = \sqrt[3]{-\frac{q}{2} - \sqrt{\Delta}} \).
     • One real solution: \( x_1 = u + v - \frac{b}{3a} \)
     • Complex solutions: \( x_2 \) and \( x_3 \) (as complex numbers).
   • For \( \Delta = 0 \):
     • Calculate \( u = \sqrt[3]{-\frac{q}{2}} \).
     • Two real solutions: \( x_1 = 2u - \frac{b}{3a} \) and \( x_2 = -u - \frac{b}{3a} \)
   • For \( \Delta < 0 \):
     • Calculate \( r = \sqrt{\frac{q^2}{4} - \Delta} \) and \( \theta = \arctan\left(\frac{\sqrt{-\Delta}}{-\frac{q}{2}}\right) \).
     • Three real solutions: \( x_1 \), \( x_2 \), and \( x_3 \) (using trigonometric functions).
6. Finalize the solutions \( x_1 \), \( x_2 \), and \( x_3 \) depending on the case and the values of \( a \), \( b \), \( c \), and \( d \).

Final Solutions:

Based on the calculation and analysis, the solutions for the given quadratic equation are:

• Solution \( x_1 \):
• Solution \( x_2 \):
• Solution \( x_3 \):