The discriminant D = $b^2$ - 4ac is the single most important quantity in quadratic equation analysis. Its sign, magnitude, and arithmetic properties together determine the complete character of the roots.

When D > 0 and is a perfect square (with a, b, c rational), roots are rational and the quadratic factors over the rationals. When D > 0 but not a perfect square, roots are irrational conjugate surds of the form (-b +/- sqrt(D))/(2a). The difference between roots is sqrt(D)/|a|, which appears frequently in JEE problems.

When D = 0, the double root x = -$\frac{b}{2a}$ is the vertex of the parabola. This case is important for tangency conditions: the line y = mx + k is tangent to y = $ax^2$ + bx + c when the resulting quadratic in x has D = 0.

For D < 0, roots are (-b +/- i*sqrt(|D|))/(2a). The modulus of each complex root is sqrt$\frac{c}{a}$, and their argument is +/- arccos(-$\frac{b}{2*sqrt(ac}$)). This modulus condition is useful in problems asking when |roots| < 1 or |roots| = 1.

Key JEE applications: (1) Finding parameter ranges for specified root nature -- set D >=, =, or < 0 and solve the resulting inequality in the parameter. (2) Discriminant of the "discriminant equation" -- when D itself is a quadratic in a parameter, analyzing when D >= 0 creates nested discriminant analysis. (3) Condition for the quadratic expression to be positive/negative definite -- requires sign of a combined with sign of D.