Quadratic Equations: JEE Main Complete Guide

Quadratic equations are both foundational and directly tested in JEE Main. As a direct topic, they contribute one to two questions per session. As a tool, they underpin complex numbers, coordinate geometry (intersection of a line and conic), and calculus (maxima-minima via setting derivative = 0). Mastering quadratic equations completely pays dividends across the entire mathematics syllabus.

Standard Form and Discriminant

Standard quadratic: ax² + bx + c = 0, a ≠ 0. Roots: x = (−b ± √D)/(2a), where discriminant D = b² − 4ac. Nature of roots: D > 0 → two distinct real roots. D = 0 → two equal real roots (each = −b/(2a)). D < 0 → two complex conjugate roots (when a, b, c are real). Sum of roots α + β = −b/a. Product of roots αβ = c/a. These Vieta's formulas are tested in every other JEE Main quadratic question — know them reflexively.

JEE Main uses Vieta's in two directions: (1) given roots, form the equation: x² − (sum)x + (product) = 0; (2) given equation, find symmetric functions of roots — like α² + β² = (α+β)² − 2αβ, or α³ + β³ = (α+β)³ − 3αβ(α+β), without finding α and β individually. Practise computing α² + β², α²β + αβ², and 1/α + 1/β using Vieta's formulas. For the complex numbers extension, see our Complex Numbers Guide.

Nature of Roots and Special Conditions

For a quadratic with rational coefficients: complex (imaginary) roots come in conjugate pairs; irrational roots come in conjugate surds. If one root is 2+√3, the other is 2−√3 (if coefficients are rational). JEE Main tests: "If one root is p+qi (complex), find the other root" — answer is p−qi. "If coefficients are rational and one root is 2+√5, find a, b, c."

Conditions for special root configurations: both roots positive → D ≥ 0, −b/a > 0, c/a > 0. Both roots negative → D ≥ 0, −b/a < 0, c/a > 0. Roots have opposite signs → c/a < 0 (product of roots negative). One root is zero → c = 0. Both roots zero → b = c = 0. Take a free mock test on algebra to practise quadratic equation problems under timed conditions.

Graph of Quadratic: Key Features

The parabola y = ax² + bx + c opens upward if a > 0, downward if a < 0. Vertex at x = −b/(2a), y = c − b²/(4a) = −D/(4a). Axis of symmetry: x = −b/(2a). The sign of the quadratic: if D < 0 and a > 0, then y > 0 for all x (always positive). If D < 0 and a < 0, then y < 0 for all x. JEE Main tests: for what values of k is the quadratic kx² + 3x + k > 0 for all x? → need D < 0 and a > 0: 9 − 4k² < 0 → k² > 9/4 → |k| > 3/2; and k > 0. So k > 3/2.

Common Roots

If two quadratics ax² + bx + c = 0 and a'x² + b'x + c' = 0 have a common root α: the common root satisfies both equations. Subtract to eliminate x²: (a−a')x² + (b−b')x + (c−c') = 0. For exactly one common root: condition is (ca'−c'a)² = (ab'−a'b)(bc'−b'c). Both roots common: a/a' = b/b' = c/c'. JEE Main presents two quadratics with parameter k and asks for k such that they share a common root.

Quadratic Inequalities

To solve ax² + bx + c > 0: find roots r₁ < r₂ (if D > 0). If a > 0: y > 0 outside the roots: x < r₁ or x > r₂. If a < 0: y > 0 between the roots: r₁ < x < r₂. Graphical approach: sketch the parabola, identify where it is above the x-axis. JEE Main uses quadratic inequalities in coordinate geometry (finding the range of k for a line to intersect a circle) and in calculus (finding where a function is increasing). For the polynomial extensions of quadratics, see our Sequences and Series Guide.

Exam Strategy

Quadratic equation questions are among the fastest to solve in JEE Main. The key is recognising the question type in 10 seconds: Is it asking about the nature of roots? (Discriminant.) Symmetric functions of roots? (Vieta's.) Sign conditions? (Graph analysis.) Common roots? (Subtraction method.) For each type, the solution path is mechanical — master the four paths and quadratic questions become guaranteed marks. For a complete algebra preparation strategy, see our Math Score 100+ Strategy. Upgrade for ₹149/month for 200+ quadratic equation problems at all JEE difficulty levels.