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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