The Quadratic Formula: Derivation, Discriminant, and Applications
The quadratic formula is one of the most celebrated equations in all of algebra. It provides the exact analytical solutions for any polynomial equation of degree 2:
ax^2 + bx + c = 0 \quad (a \ne 0)
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The Three Regimes of the Discriminant $\Delta$
The expression inside the radical $\Delta = b^2 - 4ac$ governs the fundamental algebraic nature of the roots:
- $\Delta > 0$: The square root is a positive real number. The equation has two distinct real roots, and the parabola crosses the x-axis at two distinct points.
- $\Delta = 0$: The radical vanishes. The equation has one repeated real root $x = -b / (2a)$, where the vertex is tangent to the x-axis.
- $\Delta < 0$: The radicand is negative. The equation has two complex conjugate roots $x = -\frac{b}{2a} \pm i\frac{\sqrt{|\Delta|}}{2a}$, and the graph never touches the x-axis.
Vieta's Formulas for Quadratic Equations
For any quadratic with roots $r_1$ and $r_2$, the coefficients satisfy:
r_1 + r_2 = -\frac{b}{a}
r_1 \cdot r_2 = \frac{c}{a}
Summary Comparison Table
| Discriminant Value | Nature of Roots | Parabola X-Intercepts | Example Equation |
|---|---|---|---|
| $\Delta > 0$ | 2 Distinct Real Roots | Crosses twice | $x^2 - 5x + 6 = 0$ ($x = 2, 3$) |
| $\Delta = 0$ | 1 Repeated Real Root | Tangent once at vertex | $x^2 - 6x + 9 = 0$ ($x = 3$) |
| $\Delta < 0$ | 2 Complex Conjugates | Never touches x-axis | $x^2 + 4x + 13 = 0$ ($x = -2 \pm 3i$) |