Transforming the Roots of a Quadratic Equation
If we have a known quadratic equation, we can solve it to find the roots
and
The quadratic can be written as
(1)
The roots of a quadratic are in general, complex numbers, so this equation is written in terms of
which denotes a complex number.
If we transform the roots to give new numbers
and
then these will be the roots of the quadratic
If fact we often do not need to knowand
to find the quadratic (2). We only need to know the coefficients of the equation (1), as long as the functions
and
display symmetry, so return the same numbers, possibly interchanged, ifandare interchanged.
Example: The roots of
(3)
areandFind the quadratic equation p-1 (z) with roots
and
Ifandare interchanged then so areand
so we may use the coefficients of (3)
From (1)
and
Sinceand
are roots of
we can write
Substituting the values ofandinto this expression gives
