Using the Discriminant

Date: 07/19/2000 at 00:34:33
From: Kemy
Subject: Character of a solution in the complex numbers system

Hi:

I was asked to determine the character of the solution of the 
equation:

     x^2 + 2x + 6 = 0 

in the complex number system without solving. How can I do that?

Date: 07/19/2000 at 11:37:55
From: Doctor Rob
Subject: Character of a solution in the complex numbers system

Thanks for writing to Ask Dr. Math, Kemy.

For the equation a*x^2 + b*x + c = 0, assuming a, b, and c are real 
numbers, you can tell the character of the roots by examining 
b^2 - 4*a*c, which is called the "discriminant" of the equation. If 
this is negative, then the roots are complex numbers. If this is zero, 
there are two equal real roots. If this is positive, there are two 
unequal real roots. Furthermore, if a, b, and c are rational numbers, 
then the solutions in this last case will be rational if and only if 
the discriminant is a perfect square of a rational number.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/