Ferrari’s solution of the Biquadratic.
Writing the equation …Equation A
we assume that …Equation B
Expanding and equating coefficients, we have
and
and
….Equation C
Eliminating m, n,
which reduces to
…Equation D
The second term can be removed by the substitution …Equation E
and the equation D becomes …Equation F, which is the “reducing cubic”.
Equations C become
Call the above three equations as G.
Thus, if is a root of F and
and
, the equation
can be put in the form
,
and its roots are the roots of the quadratics
.
It should be noted that the three roots of F correspond to the three ways of expressing u as the product of two quadratic factors.
Homework: Solve the equation .
Hope you enjoyed it…
More later,
Nalin Pithwa