Quadratic Formula Calculator

The quadratic formula solves any equation in the form ax² + bx + c = 0. Enter your coefficients and get the roots instantly — with the discriminant analysis showing whether solutions are real, repeated, or complex.

How to Use This Calculator

1. Enter coefficient a (must not be 0).
2. Enter coefficient b.
3. Enter coefficient c.
4. Click Calculate to see both roots (x₁ and x₂), the discriminant, and a step-by-step solution.

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

• Discriminant (D) = b² − 4ac
• D > 0: Two distinct real roots
• D = 0: One repeated real root (vertex touches x-axis)
• D < 0: Two complex conjugate roots (no real solutions)

Example Calculation

Solve: 2x² − 5x − 3 = 0 (a=2, b=−5, c=−3)

• Discriminant: (−5)² − 4(2)(−3) = 25 + 24 = 49 → D > 0, two real roots
• x = (5 ± √49) / 4 = (5 ± 7) / 4
• x₁ = (5 + 7)/4 = 3
• x₂ = (5 − 7)/4 = −0.5
• Verification: 2(9) − 5(3) − 3 = 18 − 15 − 3 = 0 ✓

Vertex Form and Parabola

The parabola y = ax² + bx + c has its vertex at x = −b/(2a). For our example: x = 5/4 = 1.25, y = 2(1.5625) − 5(1.25) − 3 = −6.125. This is the minimum point of the upward-opening parabola.

Common Mistakes to Avoid

• Forgetting to rearrange to standard form first — x² + 5 = 3x must be rewritten as x² − 3x + 5 = 0 before identifying a, b, c.
• Sign errors with negative b — In −b ± ..., if b is −5, then −b = +5. Be careful with double negatives.
• Confusing "no real solutions" with "no solutions" — A negative discriminant means two complex (imaginary) roots exist, not that the equation is unsolvable.

Frequently Asked Questions

Why use the quadratic formula instead of factoring?

Factoring only works for "nice" quadratics with integer roots. The quadratic formula works for all quadratic equations — including those with irrational or complex roots. It's always reliable.

What is completing the square?

An alternative method to solve quadratics by rewriting the equation as (x + p)² = q. The quadratic formula is actually derived by completing the square on the general form ax² + bx + c = 0.

When does a quadratic have only one solution?

When the discriminant equals zero. The parabola touches the x-axis at exactly one point — its vertex. x = −b/(2a) is the single (repeated) root.

Conclusion

The quadratic formula is one of the most important formulas in algebra — it solves every quadratic equation, guaranteed. Use this calculator for homework, exam prep, or any applied math problem involving parabolas or quadratic relationships.

Related: Scientific Calculator | Factor Calculator | Root Calculator