AlgebraHigh Yield (1-2 Qs in JEE)

Quadratic Equations Formulas

Roots of quadratic, discriminant nature, Vieta relations, condition for common roots, and location of roots.

7 Formulas•315+ Questions in Bank

Core Formulas & Mathematical Identities

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}

Roots and coefficients of ax² + bx + c = 0.

Examiner Tip: Difference of roots: |α - β| = √(D) / |a|.

D = b^2 - 4ac: \quad D > 0 \; (\text{Real, Distinct}), \quad D = 0 \; (\text{Real, Equal}), \quad D < 0 \; (\text{Complex Conjugates})

Discriminant determines the geometric intersection with the x-axis.

Examiner Tip: If a, b, c are rational and D is not a perfect square, roots occur in irrational conjugate pairs (p ± √q).

D \ge 0, \quad -\frac{b}{2a} > k, \quad a \cdot f(k) > 0

Simultaneous necessary and sufficient conditions for both roots to lie to the right of value k.

Examiner Tip: If roots lie on opposite sides of k, the ONLY condition needed is a * f(k) < 0.

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

Condition for two quadratic equations to have identical solutions.

Examiner Tip: If one equation has imaginary roots (D < 0), having ONE common root implies BOTH roots are common.

(c_1 a_2 - c_2 a_1)^2 = (a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1), \quad x = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}

Cross-multiplication determinant condition for a single shared root between two quadratics.

Examiner Tip: The common root can also be found directly by subtracting equations to eliminate the x² term.

S_n = \alpha^n + \beta^n \implies a S_n + b S_{n-1} + c S_{n-2} = 0

Powerful recurrence relation connecting powers of roots for ax² + bx + c = 0.

Examiner Tip: Very frequent in JEE Main: e.g. finding (a₁₀ - 2a₈) / 2a₉ without solving for roots explicitly.

\text{Roots } k\alpha, k\beta \implies a(x/k)^2 + b(x/k) + c = 0; \quad \text{Roots } 1/\alpha, 1/\beta \implies c x^2 + b x + a = 0; \quad \text{Roots } \alpha \pm h \implies a(x \mp h)^2 + b(x \mp h) + c = 0

Symmetric transformations of equations without recalculating individual roots.

Examiner Tip: For roots α² and β², substitute x = √y and square to eliminate radicals.

Frequently tested misconceptions in Quadratic Equations

4 Topics

›Parabolic Graph: The curve y = ax² + bx + c is a vertical parabola with axis of symmetry x = -b / (2a) and vertex at (-b / (2a), -D / (4a)).
›Concavity: Opens upward if a > 0 (has absolute minimum = -D / (4a)), opens downward if a < 0 (has absolute maximum = -D / (4a)).
›Sign of Quadratic: If D < 0, ax² + bx + c has the same sign as a for all real x ∈ ℝ. Specifically: if a > 0 and D < 0, f(x) > 0 ∀ x ∈ ℝ (strictly positive definite); if a < 0 and D < 0, f(x) < 0 ∀ x ∈ ℝ (strictly negative definite).
›Conjugate Roots: If coefficients a, b, c are rational and D is not a perfect square, irrational roots occur in conjugate pairs (p + √q and p - √q). If a, b, c are real and D < 0, imaginary roots occur in complex conjugate pairs (p + iq and p - iq).

›Sum & Product: For ax² + bx + c = 0 with roots α, β: α + β = -b/a, αβ = c/a.
›Root Difference: |α - β| = √((α + β)² - 4αβ) = √(D) / |a|.
›Symmetric Power Sums: α² + β² = (α + β)² - 2αβ; α³ + β³ = (α + β)³ - 3αβ(α + β).
›Newton's Recurrence Relation: If S_n = αⁿ + βⁿ (or p·αⁿ + q·βⁿ), then a·S_n + b·S_{n-1} + c·S_{n-2} = 0. Extremely frequent shortcut for JEE Main problems asking for expressions like (a₁₀ - 2a₈) / a₉.

›Case 1 (Both roots greater than k): D ≥ 0, -b / (2a) > k, and a·f(k) > 0.
›Case 2 (Both roots less than k): D ≥ 0, -b / (2a) < k, and a·f(k) > 0.
›Case 3 (A number k lies strictly between the roots): a·f(k) < 0. (Note: D > 0 is automatically guaranteed here; no need to evaluate D).
›Case 4 (Both roots lie within interval (k₁, k₂)): D ≥ 0, k₁ < -b / (2a) < k₂, a·f(k₁) > 0, and a·f(k₂) > 0.
›Case 5 (Exactly one root lies in interval (k₁, k₂)): f(k₁)·f(k₂) < 0 (provided neither k₁ nor k₂ is a root).

›Both Roots Common: Equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 have both roots identical if and only if a₁/a₂ = b₁/b₂ = c₁/c₂.
›Exactly One Common Root: By cross-multiplication determinant method: (c₁a₂ - c₂a₁)² = (a₁b₂ - a₂b₁)(b₁c₂ - b₂c₁). The shared common root is x = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁) = (b₁c₂ - b₂c₁) / (c₁a₂ - c₂a₁).
›Crucial JEE Trap: If one quadratic has real coefficients and D < 0 (imaginary roots), having ONE common root with another real-coefficient equation mathematically forces BOTH roots to be common, because non-real roots must occur in conjugate pairs!