📝 Quadratic Equation Solver

Find roots of quadratic equations ax² + bx + c = 0.

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What is Quadratic Equation Solver?

A quadratic equation is any equation written in the form ax² + bx + c = 0, and solving it means finding the value or values of x that make it true. These equations turn up constantly in physics problems, geometry, and competitive exam mathematics. Rather than working through the formula by hand and risking an arithmetic slip, this solver takes your three coefficients and returns the roots — real or complex — immediately.

The Quadratic Formula

For any equation ax² + bx + c = 0, the roots are found using:

x = (-b ± √(b² - 4ac)) / 2a

The expression under the square root, b² - 4ac, is called the discriminant. Its sign tells you what kind of roots to expect:

  • Discriminant > 0 → two distinct real roots
  • Discriminant = 0 → one repeated real root
  • Discriminant < 0 → two complex (imaginary) roots

Worked Example

Take x² - 5x + 6 = 0, where a = 1, b = -5, c = 6.

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1

Since the discriminant is positive, there are two real roots:

x = (5 ± √1) / 2 = (5 ± 1) / 2 → x = 3 or x = 2

You can verify this by factoring: x² - 5x + 6 = (x - 3)(x - 2), confirming the roots are 3 and 2.

Now consider x² + 2x + 5 = 0, where a = 1, b = 2, c = 5. Here the discriminant = 2² - 4(1)(5) = 4 - 20 = -16, which is negative, so the roots are complex: x = -1 ± 2i.

How to Use This Calculator

  1. Enter the coefficient of x² in the a field.
  2. Enter the coefficient of x in the b field.
  3. Enter the constant term in the c field.
  4. Click Calculate to see both roots at once.
💡 Tip: If "a" is 0, the equation is no longer quadratic — it becomes linear (bx + c = 0) — so make sure your coefficient of x² is non-zero before solving.

Need to double-check a root by raising it to a power? Our Power Calculator can help with that step.

📅 Last reviewed: July 2026 · Formulas verified against RBI/SEBI/IT Dept guidelines.