Quadratic Equations and Nature of Roots
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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.