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.

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