“Every Polynomial equation of degree
has at least on root in
”
More specifically we have, “Every Polynomial equation of degree
has
roots in
”
So, now the equation has two roots:
In a quadratic equation
with real coefficients
complex roots occur in conjugate pairs.
Example 1:
Solve:
Solution:
, Here,
Example 2:
Solve:
Here
Example 3:
Solve:
Application
(i) General Solution of the equation:
is a positive integer
, where
So, general solution of the equation , where ‘
‘ is a complex number, and
, then the ‘
‘ solutions are
where
(ii) Finding square roots of a complex number
Let , where
So, we have two roots,
Hence and
Example:
Obtain the square roots of
Solution:
Let , such that
, then
Now,
Since is non-negative, we have
Hence the square roots of are
and