Location of roots problems ask: given conditions on where the roots of $ax^2$ + bx + c = 0 lie relative to given numbers, find constraints on parameters. This is one of the most frequently tested advanced topics in JEE.

Both roots greater than k (assuming a > 0): Three conditions must hold simultaneously: (1) D >= 0 (real roots exist), (2) f(k) > 0 (k is outside the root interval), (3) -$\frac{b}{2a}$ > k (vertex is to the right of k).

Both roots less than k (a > 0): (1) D >= 0, (2) f(k) > 0, (3) -$\frac{b}{2a}$ < k.

Both roots in interval (p, q) (a > 0): (1) D >= 0, (2) f(p) > 0, (3) f(q) > 0, (4) p < -$\frac{b}{2a}$ < q.

Exactly one root in (p, q): f(p) * f(q) < 0 (by Intermediate Value Theorem). Note: D >= 0 is automatically satisfied.

k lies between the roots (a > 0): f(k) < 0. Again D >= 0 is automatic.

Roots on either side of p and q (p < q between the roots): f(p) < 0 AND f(q) < 0.

Common mistakes: (1) Forgetting the vertex condition -- f(k) > 0 alone doesn't ensure both roots exceed k; the vertex must also be correctly positioned. (2) Not considering the sign of a -- all conditions reverse when a < 0. (3) Missing D >= 0 for the "both roots in interval" case.

JEE strategy: Draw the parabola. Mark the critical point(s) k, p, q on the x-axis. Visually verify which conditions the parabola must satisfy. Then translate to algebraic inequalities.