Last Updated: April 2026
Quadratic Equations is a foundational chapter in JEE Main Mathematics, appearing in virtually every paper with 1–3 questions. Beyond its direct marks, mastery of quadratics is essential for solving problems in coordinate geometry, inequalities, complex numbers, and calculus. This guide covers every concept from standard form to reducible equations — with solved examples, formula table, and 10 MCQs.
Why Quadratic Equations is High Priority for JEE 2027
JEE Main Mathematics has 30 questions (120 marks). Quadratic Equations is part of Algebra, which historically contributes 8–12 questions per paper. Even in years where quadratics appear as a standalone topic (1 direct question), the concepts of discriminant, Vieta’s formulas, and nature of roots are applied repeatedly across the paper in other chapters.
Standard Form of a Quadratic Equation
A quadratic equation in one variable x is expressed in the standard form:
ax² + bx + c = 0
Where a, b, c are real numbers and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.
- a = coefficient of x² (leading coefficient)
- b = coefficient of x
- c = constant term
The Discriminant and Nature of Roots
The discriminant D (or Δ) determines the nature of the roots of ax² + bx + c = 0 without solving the equation:
D = b² − 4ac
| Condition | Nature of Roots | Type |
|---|---|---|
| D > 0 | Two distinct real roots | α ≠ β, both real |
| D = 0 | Two equal real roots | α = β = −b/2a |
| D < 0 | No real roots (complex conjugate pair) | α = p+qi, β = p−qi |
| D > 0, D is perfect square, a,b,c ∈ ℤ | Rational roots | Factorable |
| D > 0, D is not perfect square | Irrational roots | Conjugate surds |
Quadratic Formula
The roots of ax² + bx + c = 0 are given by:
x = (−b ± √(b² − 4ac)) / 2a
The two roots are: α = (−b + √D)/2a and β = (−b − √D)/2a
Vieta’s Formulas (Sum and Product of Roots)
For ax² + bx + c = 0 with roots α and β:
| Relation | Formula |
|---|---|
| Sum of roots | α + β = −b/a |
| Product of roots | αβ = c/a |
| Difference of roots | α − β = √D/a (when α > β) |
| Sum of squares | α² + β² = (α+β)² − 2αβ = b²/a² − 2c/a |
| Sum of cubes | α³ + β³ = (α+β)³ − 3αβ(α+β) |
Forming an equation from roots: If α and β are the roots, then: x² − (α+β)x + αβ = 0
Graph of a Quadratic Function
The quadratic function y = ax² + bx + c represents a parabola:
- If a > 0: parabola opens upward (U-shape) — minimum at vertex
- If a < 0: parabola opens downward (∩-shape) — maximum at vertex
- Vertex: (−b/2a, −D/4a)
- Axis of symmetry: x = −b/2a
- y-intercept: c (when x = 0)
- x-intercepts: the roots α and β (where D ≥ 0)
Maximum and Minimum of Quadratic Expression
For y = ax² + bx + c:
- If a > 0: Minimum value = −D/4a = (4ac − b²)/4a at x = −b/2a
- If a < 0: Maximum value = −D/4a at x = −b/2a
This is widely used in optimization problems in JEE.
Condition for Common Roots
For two quadratics a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 to have:
- One common root: (c₁a₂ − c₂a₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁)
- Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂
Equations Reducible to Quadratic Form
Many higher-degree equations can be reduced to quadratic form by substitution:
- x⁴ + px² + q = 0: Substitute t = x², gives t² + pt + q = 0
- x² + p/x² + q = 0: Substitute t = x + 1/x or x − 1/x
- a·f(x)² + b·f(x) + c = 0: Substitute t = f(x)
- Exponential form aˣ² + aˣ + c = 0: Substitute t = aˣ
Master Formula Table — Quadratic Equations
| Concept | Formula |
|---|---|
| Standard form | ax² + bx + c = 0 (a ≠ 0) |
| Discriminant | D = b² − 4ac |
| Quadratic formula | x = (−b ± √D) / 2a |
| Sum of roots (α + β) | −b/a |
| Product of roots (αβ) | c/a |
| Vertex x-coordinate | −b/2a |
| Min/Max value | −D/4a (min if a>0, max if a<0) |
| Equation from roots | x² − (α+β)x + αβ = 0 |
| α² + β² | (α+β)² − 2αβ |
| α³ + β³ | (α+β)[(α+β)² − 3αβ] |
Solved Examples
Example 1: Find the nature of roots of 2x² − 5x + 3 = 0
Solution: a = 2, b = −5, c = 3
D = b² − 4ac = (−5)² − 4(2)(3) = 25 − 24 = 1 > 0
Since D > 0 and D = 1 is a perfect square, the roots are real, distinct, and rational.
Roots: x = (5 ± 1)/4 → x = 3/2 or x = 1
Example 2: If α and β are roots of x² − 6x + 8 = 0, find α² + β²
Solution: α + β = 6 (= −b/a), αβ = 8 (= c/a)
α² + β² = (α+β)² − 2αβ = 36 − 16 = 20
Example 3: For what value of k does x² − kx + 9 = 0 have equal roots?
Solution: For equal roots, D = 0
D = k² − 4(1)(9) = 0 → k² = 36 → k = ±6
For k = 6: roots are both 3. For k = −6: roots are both −3.
Example 4: Find the maximum value of −x² + 4x − 3
Solution: a = −1, b = 4, c = −3
Maximum at x = −b/2a = −4/(2×(−1)) = 2
Maximum value = −D/4a = −(16−12)/(−4) = −4/(−4) = 1
Verification: f(2) = −4 + 8 − 3 = 1 ✓
Practice MCQs — JEE Main 2027
[cg_quiz quiz=”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”]
Frequently Asked Questions (FAQs)
Q1: How many quadratic equations questions appear in JEE Main 2027?
Quadratic Equations typically contributes 1–3 direct questions in JEE Main. However, the concepts — discriminant, Vieta’s formulas, and nature of roots — appear indirectly in 5–8 more questions across algebra, coordinate geometry, and complex numbers. So effectively, mastering this chapter helps in 8–10 questions every paper.
Q2: What is the most common mistake students make in quadratic problems?
The most frequent mistake is sign errors in Vieta’s formulas. Remember: sum of roots = −b/a (negative sign) and product = c/a (positive sign, assuming a > 0). Students often write +b/a for the sum. Also, when forming the equation from given roots, always use x² − (sum)x + (product) = 0.
Q3: How do I identify if an equation is reducible to quadratic?
Look for these patterns: (1) Even-degree polynomial like x⁴ + px² + q = 0 — substitute x² = t. (2) Reciprocal equations like x² + 1/x² + p(x + 1/x) + q = 0 — substitute x + 1/x = t. (3) Exponential equations like 4ˣ − 6·2ˣ + 8 = 0 — substitute 2ˣ = t. In every case, the key is identifying the repeated “unit” and substituting a single variable.
Q4: Is completing the square important for JEE?
Yes, completing the square is essential for: (1) deriving the quadratic formula, (2) converting y = ax² + bx + c to vertex form y = a(x−h)² + k to identify max/min quickly, and (3) solving certain integration problems in JEE Advanced. It is also used in coordinate geometry when converting conic section equations. Spend 20 minutes mastering this technique — it pays dividends across multiple chapters.
Ace JEE Mathematics at JEE Gurukul
Quadratic Equations is the cornerstone of JEE Algebra. With a strong grip on the discriminant, Vieta’s formulas, and reducible equations, you can confidently tackle not just this chapter but also inequalities, complex numbers, and sequences. Explore our complete JEE Mathematics course at JEE Gurukul Courses — structured lessons, past paper analysis, and weekly mock tests included.
JEE Tip: Spend extra time on “equations reducible to quadratic” and “conditions for common roots” — these are high-difficulty subtopics that separates toppers from the rest!