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