126 4. Equations
- (a) Verify that, if w = $&+ fi, then
(b) Solve the equation- Solve the equation
x2+18x+30=2 x2+18x+45.- (a) Without trying to solve it, explain why the equation
l-x+@%=0has no real solution.
(b) For which values of the parameter b does the equationl-x+JK%=ohave a real solution? Verify that the solution satisfies the equa-
tion.4.3 Solving Special Polynomial Equations
Since antiquity, it has been known how to solve problems which we now
recognize as quadratic equations. Lacking a convenient algebraic notation
and having no notion of imaginary number, early mathematicians gave their
solutions in the form of algorithms or geometric constructions which were
applicable only in special cases. Although some equations of higher degree
were handled by Middle Eastern mathematicians around 1000 AD, interest
in these rose markedly in the sixteenth century when Tartaglia, Cardan
and Ferrari discovered the means of solving cubic and quartic equations in
general. During the next 250 years, attempts to solve general equations of
higher degree failed, although the theory of equations was consolidated with
the help of modern notation and a number system which included surds
and imaginaries. In particular, the evidence pointed strongly towards the
proposition that every complex polynomial equation had a root and that
the number of roots, counting multiplicity, was equal to the degree.
Finally, at the outset of the nineteenth century, Ruffini and Abel estab-
lished that the roots of a general equation of degree greater than 4 could
not be expressed in terms of the coefficients as could those of equations
of lower degree. Thus it would not be possible to prove the Fundamental