Quadratic Equations

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Quadratic Equations — Key Facts for JEE Main Standard form: ax² + bx + c = 0, where a ≠ 0, a, b, c ∈ ℝ Discriminant: D = b² − 4ac determines nature of roots

• D > 0: two distinct real roots
• D = 0: two equal real roots (repeated)
• D < 0: two complex conjugate roots (no real roots) Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a Sum of roots (α + β) = −b/a; Product of roots (αβ) = c/a ⚡ Exam tip: Many JEE Main problems test the discriminant directly — always check D before deciding about roots!

---

🟡 Standard — Core Study

Standard content for students with a few days to months.

Quadratic Equations — JEE Main Study Guide

Nature of roots — sign analysis:

• Both roots positive: D ≥ 0, α + β > 0, αβ > 0, and f(0) > 0
• Both roots negative: D ≥ 0, α + β < 0, αβ > 0, and f(0) > 0
• Roots of opposite signs: αβ < 0 (sufficient condition)
• Roots reciprocal: product = 1 → c/a = 1 → c = a
• Roots equal in magnitude but opposite in sign: sum = 0 → b = 0

Common transformation techniques: If α, β are roots of ax² + bx + c = 0:

• Equation with roots (kα, kβ): a(x/k)² + b(x/k) + c = 0 → ak²x² + bkx + c = 0
• Equation with roots (α + m, β + m): substitute x − m for x → a(x−m)² + b(x−m) + c = 0
• Equation with roots 1/α, 1/β: c x² + bx + a = 0 (reverse coefficients)
• Equation with roots (α², β²): use sum = (α+β)² − 2αβ, product = (αβ)²

Common trap: After transformation, ensure the leading coefficient is non-zero. If ak² = 0, the transformed equation becomes linear!

• Key formula: D = b² − 4ac; also D must be ≥ 0 for real roots
• Common trap: For “common roots” problems (two equations share one root), if α is common root of a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0, then α satisfies both — use elimination or assume α and solve
• Exam weight: 1 question per year (4 marks); often combined with complex numbers, progression, or coordinate geometry

---

🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Quadratic Equations — Comprehensive JEE Main Notes

Condition for common roots: For equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0:

• One common root: (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁) = (a₁c₂ − a₂c₁)(c₁b₂ − c₂b₁)
• Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂

Worst-case scenario method (for parameters): When equation contains parameter m, and condition on roots is given (e.g., “roots are real and distinct”):

1. Write D ≥ 0 as inequality in m
2. Find range of m
3. Also check that coefficient of x² ≠ 0 (i.e., m value doesn’t make a = 0)
4. For sum of roots in a range, use Vieta and eliminate m

Range of quadratic expression f(x) = ax² + bx + c:

• If a > 0: minimum at x = −b/2a, minimum value = −D/4a
• If a < 0: maximum at x = −b/2a, maximum value = −D/4a
• f(x) > 0 for all x: a > 0 AND D < 0
• f(x) ≥ 0 for all x: a > 0 AND D ≤ 0
• f(x) < 0 for all x: a < 0 AND D < 0

Location of roots:

• Both roots in interval (m, n): D ≥ 0, f(m) > 0, f(n) > 0, and −b/2a ∈ (m, n)
• One root in (m, n) and one outside: f(m)·f(n) < 0
• Roots on either side of a number k: f(k) < 0

Maximum-minimum of quadratic in given domain: For x ∈ [p, q]: evaluate at endpoints and at vertex x = −b/2a if p ≤ −b/2a ≤ q

Sum of roots and product relationships: If α, β are roots of px² + qx + r = 0:

• Sum = −q/p, Product = r/p
• Sum of squares: α² + β² = (α + β)² − 2αβ = q²/p² − 2r/p
• Sum of reciprocals: 1/α + 1/β = (α + β)/αβ = −q/r
• Sum of cubes: α³ + β³ = (α + β)³ − 3αβ(α + β)

Problem-solving strategy for parametric quadratic:

1. Identify the condition on roots (real, equal, positive, etc.)
2. Express in terms of discriminant and Vieta’s formulas
3. Eliminate the parameter between conditions
4. Check domain restrictions

• Remember: “D ≥ 0 gives real roots, D < 0 gives complex roots” — always evaluate discriminant first. For “both positive roots”, use: D ≥ 0, sum > 0, product > 0, AND f(0) > 0
• Previous years: “If roots of x² − ax + 2 = 0 are α, β and α² + β² = 8, find a” [2023]; “Range of k for which kx² + (k−2)x + (k−2) > 0 for all x” [2024]

---

📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Quadratic Equations: 1 question per year, 4 marks
• Often combined with Complex Numbers (common roots) or with Arithmetic Progressions
• Pattern: given conditions on roots → find parameter value/range

💡 Pro Tips

• Vieta’s formulas (sum and product of roots) are used in most quadratic problems — master them
• For parametric questions, always use discriminant ≥ 0 as first condition
• Location of roots problems (both roots in interval) are high-scoring if you remember the conditions
• Common roots problems: use the formula or subtract the two equations and use the resulting linear relation
• Keep an eye on when coefficient of x² becomes zero — it can reduce a quadratic to linear, changing the problem entirely

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.