What is the quadratic formula?

The quadratic formula solves ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / (2a). The ± gives two roots. The expression b² − 4ac is called the discriminant (Δ). If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: two complex (imaginary) roots.

What is the discriminant?

The discriminant is Δ = b² − 4ac. It determines the nature of roots without solving the full equation: Δ > 0 → two real roots (parabola crosses x-axis twice). Δ = 0 → one real root (parabola just touches x-axis). Δ 0, so two distinct real roots (x = 3 and x = 2).

How do I find the vertex of a parabola?

For ax² + bx + c = 0, the vertex (turning point) is at x = −b / (2a) and y = a(−b/2a)² + b(−b/2a) + c. If a > 0, the parabola opens up and the vertex is the minimum. If a < 0, it opens down and the vertex is the maximum. The vertex x-coordinate is also the average of the two x-intercepts (roots).