# Quadratic Equations and Nature of Roots

> Quadratic equations in JEE Mathematics; discriminant, nature of roots, sum-product relations, common roots and location of roots with solved traps.

- Canonical URL: https://prepelephant.com/topics/jee/mathematics/quadratic-equations
- Exam / course: JEE · Subject: Mathematics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Quadratic Equations and Nature of Roots", PrepElephant, https://prepelephant.com/topics/jee/mathematics/quadratic-equations

## Direct answer

For ax^2 + bx + c = 0 with non-zero a, the discriminant D = b^2 - 4ac decides the nature of the roots: D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives a conjugate pair of non-real complex roots. The roots are x = (-b ± sqrt(D))/(2a), with sum -b/a and product c/a. These three lines drive almost every JEE question on the chapter.

## What you must remember

- Nature of roots: D > 0 real and distinct, D = 0 real and equal, D < 0 non-real complex conjugates; roots are x = (-b ± sqrt(D))/(2a). For rational coefficients, the roots are rational only when D is a perfect square.
- Sum and product: sum of roots = -b/a, product = c/a; difference of roots = sqrt(D)/|a|; alpha^2 + beta^2 = (alpha + beta)^2 - 2 alpha beta.
- Graph facts: y = ax^2 + bx + c is a parabola opening upward if a > 0 and downward if a < 0, with vertex at x = -b/(2a); the extreme value of the expression is -D/(4a).
- Sign of roots: both positive needs sum > 0, product > 0, D >= 0; opposite signs needs only product < 0.
- Location of roots: a root lies in (p, q) when f(p) and f(q) have opposite signs; both roots greater than k needs D >= 0, f(k) of the same sign as a, and -b/(2a) > k.
- Common roots: quadratics a1x^2 + b1x + c1 and a2x^2 + b2x + c2 share a root when (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1); they share both roots when a1/a2 = b1/b2 = c1/c2.
- Quadratic inequalities: factorise and apply the wavy-curve sign method; the parabola sketch settles which intervals are positive.

## Common confusion

The classic slip is concluding "both roots positive" from sum and product alone. Sum and product fix the signs only of numbers that already exist — reality is a separate demand, so D >= 0 must be added every time. Its mirror is equating "D >= 0" with "rational roots": a positive non-square discriminant still gives irrational roots — exactly the half-argument examiners dangle.

## Exam-focused takeaway

JEE Main asks discriminant classification, symmetric expressions like alpha^3 + beta^3, and wavy-curve inequalities as quick numerical-value questions. JEE Advanced pushes location of roots with a parameter (both roots inside an interval, one root on each side of a number), common-root conditions, and quadratics hidden inside functional equations. The discipline is the same at both levels: write D, write sum and product, and only then interpret.

## Frequently asked questions

### How do I know whether the roots are rational?

For rational coefficients, the roots are rational exactly when D is a perfect square; a positive non-square D gives irrational roots.

### What is the condition for both roots to be greater than a number k?

For a > 0: D >= 0, f(k) > 0 and -b/(2a) > k; mirror the signs for a < 0. All three conditions are compulsory.

### How is alpha^3 + beta^3 computed?

(alpha + beta)^3 - 3 alpha beta (alpha + beta), substituting sum = -b/a and product = c/a.

### Can a quadratic with real coefficients have one real and one non-real root?

Never. Non-real roots of real-coefficient quadratics always occur as conjugate pairs, so the count is two real or zero real.

### What does the discriminant say about the graph?

D counts the x-axis intersections of the parabola: two for D > 0, one tangential touch for D = 0, and none for D < 0.