Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Vertex: (-b/2a, f(-b/2a))
Axis of Symmetry: x = -b/2a
Nature of roots:
This calculator applies the quadratic formula to solve equations of the form ax² + bx + c = 0. It computes the discriminant (Δ = b² − 4ac) to determine if roots are real or complex, then calculates both roots using x = (−b ± √Δ) / 2a. It also finds the vertex coordinates (−b/2a, f(−b/2a)) and the axis of symmetry x = −b/2a. Results update instantly as you type.
The equation is not quadratic if a = 0 (it becomes linear). The calculator will alert you and skip the calculation.
Complex roots occur when Δ < 0. They are expressed as real ± imaginary parts (e.g., 2 + 3i). The parabola does not cross the x-axis.
Roots are where the parabola crosses the x-axis (y = 0). The vertex is the peak or trough of the parabola, found at x = −b/2a.
Quadratic equations are degree 2, so they always have exactly 2 roots (real, repeated, or complex). The parabola can intersect the x-axis at 0, 1, or 2 points.
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.