Quadratic RootsJEE Main

Concept

A quadratic $ax^2+bx+c$ graphs as a parabola. Where it crosses the x-axis are its roots. The discriminant $D = b^2-4ac$ instantly tells you whether there are two, one, or no real roots — without solving.

Key formula

Derivation

Completing the square on $ax^2+bx+c=0$ gives $\left(x+\tfrac{b}{2a}\right)^2 = \tfrac{D}{4a^2}$, hence the quadratic formula.

The sign of $D$ decides the nature: $D>0$ two real roots, $D=0$ a repeated root (parabola touches the axis), $D<0$ a complex-conjugate pair. Vieta's relations $\alpha+\beta=-b/a$ and $\alpha\beta=c/a$ link roots to coefficients directly.

Scenarios to explore

• Quadratic Roots — Discriminant, roots and the parabola's vertex.
• Equal roots: D = 0 — The parabola just touches the x-axis.
• Complex roots: D < 0 — The curve floats above the axis; the roots are a conjugate pair.
• Roots 2 and 3 — Sum 5, product 6 — the coefficients tell you both.

Real-world applications

• Projectile range and time-of-flight equations.
• Optimisation by completing the square (vertex = extremum).
• Roots of characteristic equations in physics & circuits.

JEE exam tips

• Use Vieta's formulas to build new equations whose roots are transformed (e.g. $\alpha^2,\beta^2$).
• For real roots of a downward parabola, the vertex must lie on or above the x-axis.

Common mistakes

• Sign slips in $-b/2a$ for the vertex / axis of symmetry.
• Forgetting complex roots come in conjugate pairs (for real $a,b,c$).
• Dropping the factor $4ac$ — it is $b^2 - 4ac$, not $b^2 - ac$.

Exam traps to avoid

• $D=0$ means the parabola is tangent to the x-axis (equal roots), not that it misses it.
• If $a$ and $c$ have opposite signs, $D>0$ automatically — two real roots guaranteed.