📐 Quadratic Equation Solver

Solve ax² + bx + c = 0 using the quadratic formula. Handles real and complex roots with full step-by-step solution.

Enter Coefficients: ax² + bx + c = 0

x² + x + 1 = 0
Discriminant (b² − 4ac)
0
Root x₁
—
Root x₂
—
Vertex (h, k)
—
Axis of Symmetry
x = —
Sum of Roots
—
Product of Roots
—

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

The Discriminant

The discriminant (Δ = b² − 4ac) determines the nature of the roots:

  • Δ > 0: Two distinct real roots — the parabola crosses the x-axis at two points
  • Δ = 0: One repeated real root (double root) — the parabola is tangent to the x-axis
  • Δ < 0: Two complex conjugate roots — the parabola doesn't cross the x-axis

Factored Form

When roots x₁ and x₂ are real: ax² + bx + c = a(x − x₁)(x − x₂)

What does the vertex represent? +
The vertex is the turning point of the parabola — its minimum if a > 0 (opens upward) or maximum if a < 0 (opens downward). The x-coordinate of the vertex is h = −b/(2a) and the y-coordinate is k = c − b²/(4a).
What are complex roots? +
When the discriminant is negative, the square root of a negative number is involved. The result is complex (imaginary) numbers of the form a ± bi, where i = √(−1). Complex roots always come in conjugate pairs. While they don't appear on the real number line, they have important applications in engineering and physics.