Quadratic Formula Calculator
Solve ax² + bx + c = 0 — real and complex roots with discriminant
Coefficients
ax² + bx + c = 0
a (coefficient of x²)
a
Cannot be 0 (would make it linear)
b (coefficient of x)
b
c (constant term)
c
Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant (D) = b² − 4ac
D > 0 → 2 real roots
D = 0 → 1 repeated root
D < 0 → 2 complex roots
Roots
x₁ = —
x₂ = —
Discriminant (D)—
Nature of Roots—
Vertex (h, k)—
Axis of Symmetry—
The Quadratic Formula
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. The quadratic formula x = (−b ± √D) / 2a always works, regardless of whether the equation can be factored. The discriminant D = b² − 4ac tells you the nature of roots before solving.
The vertex form of the parabola is y = a(x − h)² + k where h = −b/2a and k = f(h). The axis of symmetry is the vertical line x = h that the parabola is symmetric about.
lightbulb Example
Solve: x² − 5x + 6 = 0 (a=1, b=−5, c=6)
1D = 25 − 24 = 1 (positive → 2 real roots)
2x = (5 ± 1) / 2
✓ x₁ = 3, x₂ = 2 (factors: (x−3)(x−2))