1. Cubic Equations
7
example of such an equation with a quadratic term:
x3

3x2
 3x  1 = 0.
To solve such equations, Cardano transformed them using a generally
applicable procedure into equations of the form
y3
+ py + q = 0.
Starting with a cubic equation in the general form
x3 + ax2 + bx + c = 0,
the transformation consists in adding the summand  to the desired
solution x, which allows the quadratic and cubic terms to be com
bined:
9
/
a\3 a2 a3
/
a\3 a2
/ a\ 2 o
+ aX =(X+3)
 J *  2 7 = ( *
+
3 )  j ( *
+
3 )
+
^
a

x3
To obtain the complete transformation of the coefficients of this
equation, one replaces every occurrence of x in the equation via the
substitution
a
x = y ,
y 3 '
obtaining, after collecting terms in like powers of y, the identity
x3 + ax2 + bx + c = y3 + py + q,
with
P = ~ 3 a 2 + b'
2
3 !
L
Once one has solved the reduced cubic equation y3 + py + q = 0
with Cardano's formula, the solution of the original equation can be
obtained with the transformation x = y —  . In the concrete example
x3
—
3x2
— 3x — 1 = 0, the transformation x = y + 1 leads to the
equation
^  6j/  6 = 0,
whose solution
y
= ^ 2 + ^ 4
obtained from Cardano's formula leads to the following solution of
the original equation:
x = 1 + v^2 + v7!.