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
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 .
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