This is what Ferrari is recognized to have achieved:
Cardano was given a word problem by another mathematician, which equated to this equation:
x^{4} + 6x^{2} + 36 = 60x (original equation)
Cardano was unable to solve it. He then passed it onto Ferrari who managed to solve it. It is almost a quartic equation (ax^{4} + bx^{3} + cx^{2} + dx + e = 0) except for the fact that it does not have a ‘bx^{3}’ term in it. This makes it a depressed quartic.
Ferrari’s Solution to the Equation
Reference Structure/Equation/Step 
Applying to Cardano’s Equation 
Description 

2. x^{4} + px^{2} + qx + r = 0  x^{4} + 6x^{2} – 60x + 36 = 0  Taking 60x from both sides  
2. (x^{2} + p)^{2} = px^{2} – qx – r + p^{2}  (x^{2} + 6)^{2} = 6x^{2} + 60x – 36 + 36  Making a perfect square  
3. (x^{2} + p + y)^{2} = (2y + p)x^{2}  qx + (y^{2} + 12y)  (x^{2} + 6 + y)^{2} = (2y + 6)x^{2} + 60x + (y^{2} + 12y)  Introducing y into the perfect square. The righthand side has been made a quadratic equation in x, in the form y = ax^{2} + bx + c.  
4. (q)^{2} – 4(p + 2y)(p^{2} – r + 2py + y^{2}) = 0  3600 – 4(6 + 2y)(12y + y^{2}) = 0  Making a perfect square with y. This is done by making a discriminant zero*.  
= –8y^{3} – 120y^{2 }– 288y – 3600 = 0  Multiplying out the brackets  
ax^{3} + bx^{2} + cx + d = 0  = y^{3} – 15y^{2 } 36y + 450 = 0  Dividing equation by 8 to simplify. It is now a cubic equation*. This is known as the resolvent cubic of the original quartic.  
5. (x^{2} + 6 + y)^{2} = (2y + 6)x^{2} + 60x + (y^{2} + 12y)  (x^{2} + 6 + 4)^{2} » (2(4) + 6)x^{2} + 60x + (4^{2} + 12(4))  Solved the cubic equation and found that y » 4. Now substituting into the equation in step 2.  
(x^{2} + 10)^{2} » 14x^{2} + 60x + 64  Simplifying  
(x^{2} + 10)^{2} » (14x + 8)^{2}  Making a perfect square with the righthand side  
6.  x^{2} + 10 » ±(14x + 8)  Taking the square root of both sides  
ax^{2} + bx + c = 0  x^{2 }± 14x +10 ± 8 = 0  In quadratic form (at least enough to substitute into the quadratic formula)  
7. –b ± b^{2} – 4ac  Solve by using quadratic formula.  
x = 3.09557  If you use the positive sign  
x = 0.64608  If you use the negative sign 