Classifying solutions to quadratic equations

Consider the equation ax2 + bx + c = 0, which we wish to solve for a complex number x, and suppose that a ≠ 0. Depending on what we know about the coefficients a, b, and c, we can use the discriminant b2 − 4ac to classify the solutions even without fully calculating them.

If the coefficients are arbitrary complex numbers, then this is all that we can say:

Discriminant:Solutions:
Zero,One solution;
Nonzero,Two solutions.

If the coefficients are all real numbers, then we can say more:

Discriminant:Solutions:
Zero,One real solution;
Positive,Two real solutions;
Negative,Two conjugate imaginary solutions.

If the coefficients are all rational numbers, then we can say even more:

Discriminant:Solutions:
Zero,One rational solution;
Positive perfect square,Two rational solutions;
Positive non-square, Two conjugate irrational real solutions;
Negative,Two conjugate imaginary solutions.

If a = 1 and the coefficients are all integers, then we can say yet more:

Discriminant:Solutions:
Zero,One integer solution;
Positive perfect square,Two integer solutions;
Positive non-square, Two conjugate irrational real solutions;
Negative,Two conjugate imaginary solutions.
Go back to the course homepage.
This web page was written in 2015 by Toby Bartels, last edited on 2015 August 5. Toby reserves no legal rights to it.

The permanent URI of this web page is http://tobybartels.name/MATH-1100/2016SU/solutions/.

HTML 5