Question

Cardan's method of solving cubic equation

Answer 1

Here is your answer ☺

⭐A standard way to find a real root of a cubic equation like :

ax³ + bx² + cx + d = 0

⭐We can then find the other two roots (real or complex) by polynomial division and the quadratic formula. The solution has two steps. ⭐We first "depress" the cubic equation and then solve the depressed equation.

Hope it helps you out ⭐^_^⭐
Thanks ⭐(^^)⭐

Answer 2

Gerolamo Cardano's solutions for a cubic equation is as follows :

$f(x) = a {x}^{3} + b {x}^{2} + cx + d \\ \: \: \: \: \: \: \: \: \: \: \: \: ≡ {x}^{3} + \frac{b}{a} {x}^{2} + \frac{c}{a} x + \frac{d}{a}$

$f(x)≡(x + \frac{b}{3a} ) ^{3} + (x + \frac{b}{3a} )( \frac{3ac - {b}^{2} }{3 {a}^{2} } ) - ( \frac{9abc - 2 {b}^{3} - 27 {a}^{2} d}{27 {a}^{3} } ) = 0$

OR

$f(y)≡ {y}^{3} + py + q = 0$

Its solutions are given by :

$y_{1,2,3} = \sqrt[3]{ - ( \frac{q}{2}) + \sqrt{( \frac{q}{2}) ^{2} + ( \frac{p}{3} ) ^{3} } } + \sqrt[3]{ - ( \frac{q}{2}) - \sqrt{ {( \frac{q}{2} )}^{2} + {( \frac{p}{3} )}^{3} } }$

Thus, we can solve for any cubic equation using the cubic formula