Cubic EquationsJEE Advanced

Concept

A cubic always has at least one real root (odd degree — it goes from −∞ to +∞). Vieta ties the roots to the coefficients: sum = −b/a, pairwise sum = c/a, product = −d/a. The shape is one S-curve with up to two turning points; whether the local max/min straddle zero decides one vs three real roots.

Key formula

Derivation

Expand a(x−α)(x−β)(x−γ) and match coefficients — that's all Vieta is.

Root-counting: f'(x) = 3ax² + 2bx + c gives turning points x₁ < x₂. Three real roots ⟺ f(x₁)·f(x₂) < 0 (the hump crosses zero). Rational-root theorem: test divisors of d/a first — exams almost always plant one.

Scenarios to explore

• Cubic Equations — Roots, turning points and Vieta's formulas live.

Real-world applications

• Symmetric-function evaluations: α² + β² + γ² = (Σα)² − 2Σαβ.
• Curve sketching & area problems on cubics.
• Physics: van der Waals isotherms are cubics in V.

JEE exam tips

• α²+β²+γ² = (b² − 2ac)/a² — derive once, reuse forever.
• If roots are in AP, the middle one is −b/3a — substitute it back.
• 1/α + 1/β + 1/γ = −c/d — ratios of Vieta lines.

Common mistakes

• Sign slips in Vieta (alternating −, +, −).
• Claiming three real roots without checking the turning-value product.
• Forgetting complex roots come in conjugate PAIRS (so 1 or 3 real, never 2).

Exam traps to avoid

• x³ − 6x² + 11x − 6 = (x−1)(x−2)(x−3) — THE canonical example; recognise it.
• A cubic with two equal roots shares that root with its derivative — gcd trick.