Roots & Nature of Roots
The roots of ax² + bx + c = 0 are x = [−b ± √(b² − 4ac)] / 2a, and the quantity under the root, D = b² − 4ac, is the discriminant — it tells you the nature of the roots before you compute anything. If D > 0 there are two distinct real roots; if D = 0 the roots are real and equal (a perfect square, x = −b/2a); if D < 0 there are no real roots, just a complex conjugate pair. CAT loves to ask "for what value of k does this equation have equal/real roots?" — that is purely a discriminant condition, so set D = 0 or D ≥ 0 and solve for k, never the full formula. One more high-value fact: if a, b, c are rational and D is a perfect square, the roots are rational (the equation factorises); otherwise they are irrational and come in conjugate surd pairs p ± √q.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Solving & nature of roots
| Quadratic formula | x = [−b ± √(b² − 4ac)] / 2a |
|---|---|
| Discriminant | D = b² − 4ac |
| Two distinct real roots | D > 0 |
| Equal (repeated) real roots | D = 0 ⇒ x = −b/2a |
| No real roots (complex pair) | D < 0 |
Roots, building & extremes
| Sum of roots | α + β = −b/a |
|---|---|
| Product of roots | αβ = c/a |
| Equation from roots | x² − (α + β)x + αβ = 0 |
| Vertex (turning point) | x = −b/2a, value = −D/4a |
| Min if a > 0, Max if a < 0 | extreme value = c − b²/4a = −D/4a |