Understanding the Discriminant (Δ = b² - 4ac)
The 3 cases for determining quadratic roots:
Δ > 0 (Positive)
Two Distinct Real Roots
The parabola intersects the x-axis at two distinct points.
Δ = 0 (Zero)
One Repeated Real Root
The parabola touches the x-axis at its vertex (tangent point).
Δ < 0 (Negative)
Two Complex Conjugate Roots
The parabola never touches the x-axis (solutions involve imaginary unit i).