The Discriminant b² − 4ac

• Compute D = b² − 4ac to classify roots without fully solving.
• D > 0 → two distinct real roots; D = 0 → equal roots; D < 0 → no real roots.
• Use D in word problems on equal roots or nature of roots.

1. Discriminant — D = b² − 4ac.
2. D > 0 — two distinct real roots.
3. D = 0 — two equal real roots (repeated root).
4. D < 0 — no real roots (complex roots at higher level).
5. Condition — for equal roots: b² = 4ac; for rational roots: D is perfect square.

Find k if kx² + 6x + 3 = 0 has equal roots

Equal roots → D = 0
b² − 4ac = 0 → 36 − 12k = 0
k = 3

• Using b² + 4ac instead of b² − 4ac.
• Forgetting a when c = 0 (x² + bx = 0 still has a = 1).
• Confusing 'no real roots' with 'no solution' in all number systems.

• If D = 25 for a quadratic, how many distinct real roots?
• For what k does x² + kx + 9 = 0 have equal roots?
• Can D be negative for x² + 1 = 0?

Open the Practice tab for graded questions on The Discriminant b² − 4ac.