Numarly

Quadratic Equation Calculator

Solve equations of the form ax² + bx + c = 0

What is it?

This calculator solves quadratic equations of the form ax² + bx + c = 0, finding both roots (solutions) using the quadratic formula, and tells you whether the roots are real or complex based on the discriminant.

Formula

x = (−b ± √(b² − 4ac)) / 2a

Formula Explanation

The discriminant (b² − 4ac) inside the square root determines how many real solutions exist: positive gives two distinct real roots, zero gives one repeated real root (the parabola just touches the x-axis), and negative means the parabola never crosses the x-axis, giving two complex roots instead.

Example Calculation

For x² − 5x + 6 = 0 (a=1, b=−5, c=6): D = 25 − 24 = 1, so roots are x₁ = 3 and x₂ = 2.

How to Use

  1. Enter the coefficients a, b, and c from your equation.
  2. Click Solve Equation.
  3. View the discriminant, nature of the roots, both solutions (x₁ and x₂), and the parabola graph.

Benefits

  • Handles all three root cases: real & distinct, real & equal, and complex.
  • Visualizes the parabola with root markers for real solutions.
  • Shows the discriminant so you understand why the roots turned out the way they did.

Use Cases

  • Solving algebra homework or exam problems.
  • Physics problems involving projectile motion (a parabolic trajectory).
  • Optimization problems where a parabola models cost, profit, or area.

What Your Result Means

The roots (x₁, x₂) are the x-values where the parabola y = ax² + bx + c crosses (or touches) the x-axis. For complex roots, the parabola never touches the x-axis at all — the graph stays entirely above or below it.

Tips

  • Check the discriminant sign first to know what kind of roots to expect before solving.
  • If a is negative, the parabola opens downward instead of upward.
  • Use the graph to visually confirm where the curve crosses zero.

Common Mistakes

  • Entering a = 0, which turns the equation linear rather than quadratic.
  • Forgetting the ± sign, which produces only one of the two roots.
  • Mixing up b² with (b)² when b is negative — squaring always gives a positive result.

FAQs

What does the discriminant tell me?

The discriminant (b² − 4ac) determines the nature of the roots: positive means two real roots, zero means one repeated real root, and negative means two complex roots.

Why must a not be zero?

If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic — use the Linear Equation Solver for that case instead.

What are complex roots?

When the discriminant is negative, the roots involve the imaginary unit i (√-1) — these are valid mathematical solutions but not real numbers.

What does the graph show?

The graph plots y = ax² + bx + c around its vertex, with red markers showing where the curve crosses the x-axis (the real roots), if any.

Can a quadratic equation have only one solution?

Yes, when the discriminant equals zero, both roots are equal, meaning the parabola touches the x-axis at exactly one point.

Complex roots are shown in the form real ± imaginary·i, but are not plotted on the graph.

Last updated: July 25, 2026