Lesson Solution of the quadratic equation with real coefficients on complex domain

Solution of the quadratic equation with real coefficients on complex domain

In this lesson we consider the solution of the quadratic equation
,              (1)
with real coefficients , and on complex domain.
This means that we allow and look for roots among the set of complex numbers.

Let me remind you how to solve of the quadratic equation with real coefficients on real domain.
This issue was considered in the lesson Introduction into Quadratic Equations of the module Quadratic Equation of this site,
as well as in other lessons in that module.

The key formula for the solution of the quadratic equation is the quadratic formula
.      (2)
This formula was deduced in the lesson PROOF of quadratic formula by completing the square of the module Quadratic Equation of this site.
The method "completing the square" was used in this deduction.

The discriminant

plays the key role in solution of the quadratic equation.

If the discriminant value is positive then the equation has two real roots
.                   (3)

If the discriminant value is equal to zero then the equation has only one real root
.                      (4)

If the discriminant value is negative then the equation has no real roots.

Below are the major points and details of the solution of the quadratic equation (1) with real coefficients on complex domain (on the set of complex numbers).

1. The same method "completing the square" works for complex roots also.
    The method produces the same quadratic formula (2), which is valid for complex roots as well.

2. In contrast with the case of real domain, the quadratic equation (1) with real coefficients always has roots on the complex domain.

3. It has exactly two roots in the complex domain, and the quadratic formula (2) gives their values.

4. If the discriminant value is positive then these complex roots are actually real numbers.

5. If the discriminant value is equal to zero then the formula (2) actually reduces to formula (4).
    It produces real roots. Two roots actually merge into one root in this case.

6. If the discriminant value is negative then the equation has two complex roots given by formula (3).
    These complex roots are not real numbers. They are conjugate complex numbers.

You can always represent the quadratic formula in the form
.
If the discriminant value is negative real number, you can re-write it as
.             (5)

It was shown in the lesson How to take a square root of a complex number that
= +-
(see the Example 5 of that lesson).

Using this, you can re-write formula (5) in other form to represent complex roots more explicitly:
,              (6a)

or, which is the same, in the form
.     (6b)

Example 1. Solve quadratic equation on complex domain.
Solution.
Calculate the discriminant: .
The discriminant is positive real number.
Hence, the equation has two real roots
,
and
.

Answer. The equation has two real roots and .

Example 2. Solve quadratic equation on complex domain.
Solution.
Calculate the discriminant: .
The discriminant is equal to zero.
Hence, the equation has only one real root
.

Answer. The equation has only one real root .

Example 3. Solve quadratic equation on complex domain.
Solution.
Calculate the discriminant: .
The discriminant is negative real number.
Hence, the equation has two complex roots
,
and
.



For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane
Addition and subtraction of complex numbers in complex plane
Multiplication and division of complex numbers in complex plane
Raising a complex number to an integer power
How to take a root of a complex number
Solution of the quadratic equation with real coefficients on complex domain      (this lesson)
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain