Comprehensive Guide to the Quadratic Formula & Equations
A quadratic equation is a second-order polynomial equation of the canonical form ax² + bx + c = 0, where a ≠ 0. Quadratic functions describe parabolic trajectories in classical projectile physics, revenue optimization curves in microeconomics, and wave harmonics in electronics.
x = [ -b ± √(b² - 4ac) ] / (2a)
Understanding the Discriminant (Δ)
The expression underneath the square root, Δ = b² - 4ac, is known as the discriminant. It determines the geometric relationship between the parabola and the x-axis:
- Δ > 0 (Positive): The parabola intersects the x-axis at two distinct real coordinates \(x_1\) and \(x_2\).
- Δ = 0 (Zero): The vertex of the parabola touches the x-axis at exactly one point (one repeated real root \(x = -b / (2a)\)).
- Δ < 0 (Negative): The parabola never crosses the x-axis. The equation yields two complex conjugate roots: \(x = p \pm qi\), where \(i = \sqrt{-1}\).
Finding the Vertex and Axis of Symmetry
Every quadratic graph forms a symmetric parabola. The line of symmetry runs vertically through the vertex:
- Axis of Symmetry: \(x = h = -\frac{b}{2a}\)
- Vertex Coordinates: \((h, k)\) where \(k = c - \frac{b^2}{4a}\)
- Extrema: If \(a > 0\), the vertex represents the global minimum. If \(a < 0\), the vertex represents the global maximum.
Frequently Asked Questions
Why cannot coefficient 'a' be zero?
If \(a = 0\), the \(x^2\) term disappears and the equation degrades into a linear equation \(bx + c = 0\), which only has one root \(x = -c / b\).
How do you solve quadratic equations with complex roots?
When \(\Delta < 0\), write \(\sqrt{\Delta} = \sqrt{-|\Delta|} = i\sqrt{|\Delta|}\). Then the solutions are \(x = \frac{-b \pm i\sqrt{|\Delta|}}{2a}\).