JEE Mains 2026 Revision Capsule: Quadratic Equations
Quadratic equations form the backbone of JEE algebra. The exam rarely asks direct formulas; instead, it tests discriminant tricks, root nature, parameter-based questions, and graphical understanding. This revision capsule gives all high-yield concepts in a crisp format.
1. Standard Form
A quadratic equation: $$ ax^2 + bx + c = 0,\ a \ne 0 $$
2. Roots Formula
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
3. Discriminant & Nature of Roots
$$ D = b^2 - 4ac $$
- $D > 0$ → real & distinct
- $D = 0$ → real & equal
- $D < 0$ → complex conjugate roots
JEE Trick: Many problems hide $D$ inside expressions like $p^2 - 4q$ or ask for values of parameters where roots change nature.
4. Sum & Product of Roots
If roots are $\alpha, \beta$:
$$ \alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a} $$
Useful JEE fact: If roots lie on the same side of origin → check signs of sum and product.
5. Condition for Real Roots in Terms of Vertex
Vertex at $$ x = -\frac{b}{2a} $$
Minimum/maximum value: $$ f_{min/max} = -\frac{D}{4a} $$
Key JEE idea: For $ax^2+bx+c \ge 0$ or $\le 0$, directly test the sign of $a$ and $D$ using the above result.
6. Roots in a Given Interval
To check if both roots lie between $p$ and $q$:
$$ f(p)\cdot f(q) > 0 \quad \text{and} \quad -\frac{b}{a} \in (p+q) $$
This appears frequently in JEE when checking bounded roots.
7. Graphical Insight
• $a > 0$ → parabola opens upward • $a < 0$ → opens downward • Intersection with x-axis shows roots
Zero intersections → no real roots → $D < 0$.
8. JEE Traps (Be Careful!)
Trap 1: Hidden Quadratic in Disguise — Rational expressions often reduce to quadratic after clearing denominator.
Trap 2: Parameter shifting — If $a,b,c$ depend on $k$, roots change nature as $D(k)$ crosses zero.
Trap 3: Greatest and smallest values — Always rewrite quadratic in vertex form.
9. Quick JEE-Style Examples
Q1. For what values of $k$ does the equation $$ x^2 - (k+3)x + k = 0 $$ have equal roots?
Solution: $D=0$ → $$ (k+3)^2 - 4k = 0 $$ $$ k^2 + 2k + 9 = 0 $$ No real solution → No real value of k gives equal roots.
Q2. If the roots lie on the same side of origin, what must be true?
Condition: $$ \alpha\beta > 0 \quad \text{and} \quad \alpha+\beta > 0 \ (\text{both positive}) $$ or $$ \alpha\beta > 0 \quad \text{and} \quad \alpha+\beta < 0 \ (\text{both negative}) $$
10. Killer Shot Formula Set (Ultra-Crisp)
$$ x = \frac{-b\pm\sqrt{D}}{2a} $$ $$ \alpha+\beta = -\frac{b}{a} $$ $$ \alpha\beta = \frac{c}{a} $$ $$ D = b^2 - 4ac $$ $$ f_{min/max} = -\frac{D}{4a} $$ $$ x_{vertex} = -\frac{b}{2a} $$
This capsule is part of the StudyBeacon JEE Mains 2026 Revision Series — fast, focused, and scientifically structured for maximum recall.
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