Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Calculates real and complex roots, discriminant, vertex, and axis of symmetry.
Step-by-step Solution
- Identify: a = 1, b = -5, c = 6
- Compute discriminant: Δ = b² − 4ac = -5² − 4×1×6 = 1
- √Δ = √1 = 1
- x₁ = (−-5 + 1) / (2×1) = 3
- x₂ = (−-5 − 1) / (2×1) = 2
- Vertex: h = −b/(2a) = 2.5, k = -0.25
The Quadratic Formula
For ax² + bx + c = 0, the solutions are:
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant Δ = b² − 4ac determines the nature of the roots:
- Δ > 0 — Two distinct real roots
- Δ = 0 — One repeated real root (the parabola touches the x-axis)
- Δ < 0 — Two complex conjugate roots (no real solution)
The vertex of the parabola is at h = −b/(2a), k = c − b²/(4a).
Frequently Asked Questions
x = (−b ± √(b² − 4ac)) / (2a). It gives the values of x where ax² + bx + c = 0.
The discriminant Δ = b² − 4ac tells us how many real solutions exist. Δ > 0: two real roots. Δ = 0: one repeated real root. Δ < 0: two complex (imaginary) roots.
The vertex is the highest or lowest point of the parabola. Its x-coordinate is h = −b/(2a) and y-coordinate is k = c − b²/(4a).
When a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.
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