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Quadratic Formula Calculator – Solve ax² + bx + c = 0

Quadratic Formula Solver

Equation: ax² + bx + c = 0

The Quadratic Formula Calculator solves any quadratic equation of the form ax² + bx + c = 0. It computes the discriminant, finds real or complex roots, and provides a complete step‑by‑step explanation. The quadratic formula is one of the most important tools in algebra, used in physics, engineering, finance, and many other fields. Whether you need to find the x‑intercepts of a parabola, solve projectile motion problems, or optimise profit functions, this calculator gives instant, accurate results.

Quadratic ParabolaxyVertexThe x-intercepts are the roots

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

The Discriminant

Δ = b² – 4ac

• If Δ gretaer than 0 → two distinct real roots

• If Δ = 0 → one real double root (vertex touches x-axis)

• If Δ less than 0 → two complex conjugate roots (no real x-intercepts)

Real‑World Applications

  • Physics: Projectile motion (height vs. time), free‑fall problems.
  • Engineering: Stress‑strain analysis, optimisation of structures.
  • Economics: Profit maximisation, break‑even analysis.
  • Geometry: Finding dimensions of shapes given area or perimeter constraints.
Derivation of the Quadratic Formula

The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. Divide by a, move the constant term, add (b/2a)² to both sides, factor the left, then take the square root. The result is the familiar formula. This derivation works for all a, b, c (a ≠ 0).

Common Mistakes When Using the Quadratic Formula

  • Forgetting the ± sign: Both roots must be considered.
  • Mis‑identifying signs of b and c: Write the equation in standard form first (ax² + bx + c = 0).
  • Incorrect calculation of discriminant: b² – 4ac, not b² – 4a + c.
  • Dividing by a only after subtracting c: The formula divides by 2a at the end.

How to Derive the Quadratic Formula (Step‑by‑Step)

  1. Start with ax² + bx + c = 0 (a ≠ 0).
  2. Divide both sides by a: x² + (b/a)x + c/a = 0.
  3. Move constant term: x² + (b/a)x = -c/a.
  4. Complete the square: add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)².
  5. Left side is (x + b/2a)² = (b² - 4ac)/4a².
  6. Take square root: x + b/2a = ±√(b² - 4ac) / (2a).
  7. Solve for x: x = [-b ± √(b² - 4ac)] / (2a).

Use this quadratic formula calculator to check your homework, verify your work, or explore how changing coefficients affects the roots. The step‑by‑step output makes it an excellent learning tool for students.

Step‑by‑Step Manual Example

Solve x² – 5x + 6 = 0:

Identify a = 1, b = -5, c = 6

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

√Δ = 1

x₁ = (5 – 1) / (2×1) = 4/2 = 2

x₂ = (5 + 1) / 2 = 6/2 = 3

Roots: x = 2, 3

Frequently Asked Questions about Quadratic Equations

What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). It solves any quadratic equation ax² + bx + c = 0.
What is the discriminant?
The discriminant Δ = b² - 4ac determines the nature of roots: Δ > 0 → two real roots; Δ = 0 → one real double root; Δ < 0 → two complex conjugate roots.
Can I use decimals for coefficients?
Yes, the calculator accepts decimal values like 1.5, -2.3, etc. It will compute accurate roots.
What if a = 0?
If a = 0, the equation is not quadratic (it is linear). Our calculator will show an error message.