Quadratic Equation Calculator

Solve ax² + bx + c = 0 instantly. Get real & complex roots, discriminant, vertex, and interactive graph. Perfect for students and teachers.

Standard Form: ax² + bx + c = 0    (a ≠ 0 for quadratic)

Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a)

x² - 5x + 6 = 0
x² + 2x + 1 = 0
x² + x + 1 = 0
2x² - 4x + 2 = 0
-x² + 4x - 3 = 0
x² - 4 = 0
Quadratic Equation Analysis
1. Equation
f(x) = 1x² - 3x + 2
2. Discriminant (Δ = b² - 4ac)
Δ = 1
Two distinct real roots
3. Roots (x-intercepts)
x₁ =
2.00
x₂ =
1.00
4. Vertex (turning point)
Vertex = (1.50, -0.25)
Axis of symmetry: x = 1.50
Concave up (a > 0)

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0 (with a ≠ 0). Its graph is a parabola.

Key Components:

  • Discriminant (Δ): b² - 4ac — determines the nature of roots.
    • Δ > 0 → two distinct real roots
    • Δ = 0 → one real root (double root)
    • Δ < 0 → two complex conjugate roots
  • Vertex: The minimum or maximum point of the parabola. x_v = -b/(2a), y_v = f(x_v).
  • Axis of symmetry: Vertical line x = -b/(2a).
  • Roots (solutions): x = [-b ± √Δ] / (2a).

Forms of Quadratic Equations

Form Equation Useful for
Standard ax² + bx + c = 0 Identifying coefficients
Vertex y = a(x - h)² + k Vertex (h,k) directly
Factored a(x - r₁)(x - r₂) Roots r₁, r₂

Properties of Parabolas

1
Opening direction: If a > 0, parabola opens upward (vertex is minimum). If a < 0, opens downward (vertex is maximum).
2
Width: |a| larger → narrower parabola; |a| smaller → wider.
3
y-intercept: Always at (0, c).

Applications

  • Physics: Projectile motion (height vs. time)
  • Economics: Profit maximization (quadratic revenue/cost)
  • Geometry: Area optimization problems
  • Engineering: Parabolic reflectors and bridges

Calculator Features:

  • Real-time updates as you type (debounced)
  • Handles a = 0 gracefully (linear equation mode)
  • Complex roots shown in standard a+bi form, plus exact surd form when possible
  • Interactive graph with roots & vertex highlighted
  • Automatic range selection to frame the parabola

Frequently Asked Questions

If a = 0, the equation becomes linear (bx + c = 0). The calculator will solve it as a linear equation: one root x = -c/b (if b ≠ 0). If both a and b are zero, it's either constant (c=0: infinite solutions) or contradiction (c≠0: no solution).

When discriminant is negative, roots are complex conjugates. They are shown as "α ± βi" where α = -b/(2a) and β = √(-Δ)/(2a). For integer/rational coefficients, an exact surd form (e.g., -1/2 ± √3/2 i) is also provided below the decimal approximation.

The vertex is the highest or lowest point of the parabola. In real-world problems, it often represents the maximum (e.g., projectile height) or minimum (e.g., cost) value.

Yes! All coefficient inputs accept decimal numbers. For fractions, you can enter them as decimals (e.g., 0.5 for 1/2). The calculator works with floating-point arithmetic.

In "Auto" mode, the range is set to [vertex_x - 4, vertex_x + 4] extended to include roots and y-intercept, ensuring the parabola's key features are visible. You can also choose fixed ranges or set custom min/max.