Quadratic Equation Calculator — Solve ax² + bx + c = 0

Instant solutions with steps, discriminant, and exact/decimal roots. Private by design—everything runs locally in your browser.

Coefficients & Actions

Tips: Ctrl/Cmd + K focuses a. Ctrl/Cmd + Enter solves again.

Solution

Result: Enter a, b, c and click “Solve Equation”.

Calculation Steps

Steps will appear here.

Understanding Quadratic Equations

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0 where a, b, and c are real numbers and a ≠ 0. Quadratics appear everywhere: projectile motion, optimization problems, area/geometry questions, and many modeling tasks in science and engineering.

The Quadratic Formula

Every quadratic can be solved using the quadratic formula:

x = [-b ± √(b2 - 4ac)] / (2a)

The expression under the square root, b² − 4ac, is called the discriminant (Δ). It determines the nature of the roots:

  • If Δ > 0 → two distinct real roots.
  • If Δ = 0 → one real repeated root.
  • If Δ < 0 → two complex conjugate roots.

How to use this calculator

  1. Enter the coefficients a, b, and c for ax² + bx + c = 0.
  2. Click Solve Equation to compute the discriminant and roots.
  3. Review the step-by-step working to see substitutions, Δ, and final solutions.

Everything runs client-side in your browser, so your inputs never leave your device.

Worked example

Suppose a = 1, b = −3, c = 2. Then Δ = (−3)² − 4·1·2 = 9 − 8 = 1. Since Δ > 0, there are two real roots:

x1 = [−(−3) + √1] / (2·1) = (3 + 1) / 2 = 2
x2 = [−(−3) − √1] / (2·1) = (3 − 1) / 2 = 1

Common mistakes to avoid

  • Forgetting that a must not be zero; if a = 0, the equation is linear.
  • Dropping the ± sign and only computing one root.
  • Mishandling the square of a negative number (e.g., (−b)² = b², not −b²).
  • Not simplifying fractions at the end (divide the entire numerator by 2a).

5 Fun Facts about Quadratics

The formula is ancient

Babylonian tablets (circa 2000 BCE) already had quadratic recipes—long before symbolic algebra or the letter “x.”

Math history

Vertex at −b/2a

The parabola’s peak/trough sits at x = −b/(2a). One quick division reveals symmetry and the axis of the curve.

Graph shortcut

Roots sum & product

For roots r₁, r₂: r₁ + r₂ = −b/a and r₁·r₂ = c/a. Vieta’s formulas let you sanity-check without re-solving.

Built-in check

Completing the square born here

The quadratic formula is derived by completing the square—same move that converts ax²+bx+c to vertex form.

Derivation link

Complex roots are twins

When Δ < 0, roots arrive as conjugate pairs \(p \pm qi\). Their real parts still add to −b/a and multiply to c/a.

Symmetry rules

Quadratic Equation: FAQs

What is the quadratic formula?

x = [-b ± √(b² − 4ac)] / (2a). It solves ax² + bx + c = 0 for x when a ≠ 0.

What does the discriminant (Δ) tell me?

Δ = b² − 4ac. If Δ > 0: two real roots; Δ = 0: one repeated real root; Δ < 0: two complex roots.

Can this calculator show complex roots?

Yes. When Δ < 0, roots are shown in a ± bi form (real and imaginary parts).

Is my data private?

Yes. All calculations are done locally in your browser; no data is uploaded.

Explore more tools