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Quadratic Equation Calculator

Solve any equation of the form ax² + bx + c = 0 instantly. Get real or complex roots, the discriminant, vertex coordinates, axis of symmetry, and a live parabola graph.

Real & Complex
Root Support
Live
Parabola Graph
100%
Free & Accurate

Why Use This Calculator?

  • ✓ Handles real and complex (imaginary) roots automatically.
  • ✓ Shows the discriminant, vertex, axis of symmetry, and y-intercept.
  • ✓ Draws a live graph of the parabola with roots and vertex marked.

On This Page

Quadratic Calculator Discriminant Reference Formulas Worked Examples Best Practices FAQs Related Calculators

Quadratic Equation Calculator

ax² + bx + c = 0

Enter the coefficients a, b, and c.

a cannot be 0 — otherwise the equation is linear, not quadratic. Use decimals or negative numbers freely.

Parabola Graph

Solution

Equation
x² − 3x + 2 = 0
Discriminant (Δ)
1
Nature of Roots
Two Real Roots
Roots
x₁ = 2, x₂ = 1
Vertex
(1.5, −0.25)
Axis of Symmetry
x = 1.5
Sum of Roots (−b/a)
3
Product of Roots (c/a)
2
Y-Intercept
(0, 2)

Discriminant Quick Reference

Discriminant (b² − 4ac) Nature of Roots Graph Behavior
Greater than 0Two distinct real rootsParabola crosses the x-axis twice
Equal to 0One repeated real rootParabola touches the x-axis once (vertex on axis)
Less than 0Two complex conjugate rootsParabola does not touch the x-axis

Quadratic Equation Formulas

Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ (2a)

Discriminant

Δ = b² − 4ac

Vertex Coordinates

h = −b ÷ (2a),   k = f(h)

Axis of Symmetry

x = −b ÷ (2a)

Sum & Product of Roots

Sum = −b ÷ a,   Product = c ÷ a

Y-Intercept

Set x = 0 → y = c

Worked Examples

Example 1 — Solve x² − 3x + 2 = 0

Δ = (−3)² − 4(1)(2) = 1 → x = (3 ± 1) ÷ 2 → x = 2 or x = 1

Example 2 — Solve x² + 4x + 4 = 0

Δ = 16 − 16 = 0 → x = −2 (repeated root)

Example 3 — Solve x² + x + 1 = 0

Δ = 1 − 4 = −3 → x = −0.5 ± 0.866i (complex roots)

Example 4 — Vertex of 2x² − 8x + 3 = 0

h = 8 ÷ 4 = 2, k = f(2) = 2(4) − 16 + 3 = (2, −5)

Where Quadratic Equations Are Used

🚀 Physics & Motion

Model projectile motion, falling objects, and trajectories using position-time equations.

🏗 Engineering

Solve structural, electrical, and optimization problems that reduce to quadratic form.

💰 Finance & Economics

Model profit, revenue, and cost functions to find maximum or minimum values.

🎓 Algebra & Exams

Solve textbook problems, standardized test questions, and factoring exercises.

📐 Geometry

Find dimensions of shapes when area or perimeter relationships form a quadratic.

💻 Computer Graphics

Calculate curve intersections and parabolic paths in simulations and games.

Best Practices & Common Mistakes

✅ Best Practices

  • Always check the discriminant before solving for roots.
  • Write the equation in standard form (ax² + bx + c = 0) first.
  • Keep track of signs carefully when substituting into the formula.
  • Use the vertex to quickly find the maximum or minimum value.
  • Verify roots by substituting them back into the original equation.

❌ Common Mistakes

  • Forgetting the ± sign, which produces only one root.
  • Using a = 0, which turns the equation linear instead of quadratic.
  • Dropping the negative sign inside the square root when Δ is negative.
  • Confusing the axis of symmetry with a root of the equation.
  • Rounding intermediate steps too early, causing inaccurate final roots.

Frequently Asked Questions

The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ (2a), used to solve any equation of the form ax² + bx + c = 0.
The discriminant, b² − 4ac, reveals the nature of the roots: positive means two real roots, zero means one repeated real root, and negative means two complex roots.
Yes. When the discriminant is negative, the equation has two complex (imaginary) roots instead of real number solutions, and the parabola never crosses the x-axis.
The vertex x-coordinate is h = −b ÷ (2a), and the y-coordinate k is found by substituting h back into the original equation.
If a equals 0, the equation is no longer quadratic — it becomes a linear equation (bx + c = 0) and should be solved with linear methods instead.
For ax² + bx + c = 0, the sum of the roots equals −b ÷ a and the product of the roots equals c ÷ a.
Yes. It uses the standard quadratic formula and handles real, repeated, and complex roots with full decimal precision.

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