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Quadratic Formula Calculator

Solve quadratic equations of the form ax² + bx + c = 0. Shows roots, discriminant, and vertex of the parabola.

FREE COMPLEX ROOTS
ax² + bx + c = 0
Discriminant (b²-4ac)
Vertex

📐 The Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a
Discriminant (Δ = b² - 4ac)
  • Δ > 0 — Two distinct real roots
  • Δ = 0 — One repeated (double) root
  • Δ < 0 — Two complex conjugate roots

💡 Examples

x² - 5x + 6 = 0 (a=1, b=-5, c=6)

x₁ = 3, x₂ = 2

Δ = 25 - 24 = 1 > 0 (two real roots)

x² + 1 = 0 (a=1, b=0, c=1)

x₁ = 0 + 1i, x₂ = 0 - 1i

Δ = 0 - 4 = -4 < 0 (complex roots)

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💡 Quick Tip

The vertex of the parabola is at x = -b/(2a). If a > 0, the parabola opens upward; if a < 0, it opens downward.