For real coefficients a, b, c with a != 0:
| D value | Root nature |
|---|---|
| D > 0 | Two distinct real roots |
| D > 0, D is perfect square, a,b,c rational | Two distinct rational roots |
| D > 0, D not perfect square | Two irrational roots (conjugate surds if a,b,c rational) |
| D = 0 | One repeated real root = - |
| D < 0 | Two complex conjugate roots |
Additional facts:
- If a, b, c are rational and one root is p + sqrt(q), the other is p - sqrt(q)
- If a, b, c are real and one root is p + iq, the other is p - iq
- If a + b + c = 0, then x = 1 is a root (quick check!)
- If a - b + c = 0, then x = -1 is a root