Roots (x₁, x₂) · Discriminant (Δ) · Parabola Graph · Vertex (h, k)

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 with real or complex roots, vertex coordinates, and interactive parabola graphing.

Presets:

Standard Quadratic Form

1x² + (-5)x + (6) = 0

Coefficient a (x²)Must be non-zero

Coefficient b (x)Linear term

Constant cConstant term

Calculated Roots

x₁ = 3

x₂ = 2

Discriminant (Δ = b² - 4ac)

Δ > 0 (Two real roots)

Parabola Vertex (h, k)

(2.5, -0.25)

Global Minimum (Opens Up)

Axis & Y-Intercept

Axis: x = 2.5

Y-int: (0, 6)

Vertical axis of symmetry

Interactive Parabola Visualizationy = ax² + bx + c

Vertex (2.5, -0.25) Roots (x-intercepts) Axis x = 2.5

Alternative Algebraic Forms

Vertex Form a(x - h)² + k:

y = (x - 2.5)² - 0.25

Factored Form a(x - r₁)(x - r₂):

y = (x - 3)(x - 2)

General Quadratic Formula:

x = (-b ± √(b² - 4ac)) / (2a)

1. Identify coefficients: a = 1, b = -5, c = 6

2. Compute Discriminant Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

3. Since Δ > 0, there are two distinct real roots.

4. x = (-b ± √Δ) / 2a = (-(-5) ± √1) / (2 × 1)

x₁ = (5 + 1) / 2 = 3

x₂ = (5 - 1) / 2 = 2

Understanding Quadratic Functions & Formula

The discriminant reveals the number and nature of roots without needing to compute the square root:

Δ > 0: Two distinct real roots where the parabola crosses the x-axis twice.
Δ = 0: Exactly one repeated real root (the parabola touches the x-axis at its vertex).
Δ < 0: Two complex conjugate roots (the parabola never intersects the x-axis).

The vertex (h, k) represents the highest (maximum) or lowest (minimum) point of the parabolic trajectory:

h = -b / (2a)
k = c - (b² / 4a)

If a > 0, the parabola opens upward with a minimum at k. If a < 0, it opens downward with a maximum.

The formula is derived by completing the square on the standard polynomial ax² + bx + c = 0: