Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 with step-by-step working.
ax² + bx + c = 0
Discriminant (Δ = b² − 4ac)
x₁
x₂
No real solutions — complex roots
How It Works
Enter the coefficients a, b, and c for the equation ax² + bx + c = 0. The solver applies the quadratic formula and shows real roots (or complex roots if applicable) with full working.
**Quadratic Equation Solver — Solve ax² + bx + c = 0**
Quadratic equations are one of the most fundamental topics in algebra. They appear in physics (projectile motion), economics (profit maximisation), geometry (area problems), and engineering. Our solver uses the quadratic formula and shows every step.
**The Quadratic Formula**
For ax² + bx + c = 0:
x = (-b ± √(b² – 4ac)) / 2a
**The Discriminant**
The discriminant D = b² – 4ac determines the nature of roots:
| Discriminant | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root (double root) |
| D < 0 | Two complex conjugate roots (no real solutions) |
**Step-by-Step Example**
Solve: 2x² – 7x + 3 = 0
1. Identify: a=2, b=-7, c=3
2. Discriminant: D = (-7)² – 4(2)(3) = 49 – 24 = 25
3. √D = 5
4. x₁ = (7 + 5) / 4 = 3
5. x₂ = (7 – 5) / 4 = 0.5
**Alternative Solution Methods**
**Factoring:** Find factors of ac that add to b
**Completing the Square:** Rewrite as (x + p)² = q
**Graphing:** The roots are where the parabola crosses the x-axis
**Vertex Form**
Every quadratic can be written as: f(x) = a(x – h)² + k
Where (h, k) is the vertex of the parabola:
- h = -b / 2a
- k = c – b²/4a
Our solver also shows the vertex coordinates and whether the parabola opens upward (a > 0) or downward (a < 0).
**Real-World Applications**
*Physics* — Calculating when a projectile hits the ground (height = -4.9t² + v₀t + h₀)
*Business* — Revenue maximisation (quadratic profit functions)
*Engineering* — Beam deflection and structural analysis
*Geometry* — Finding dimensions of rectangles with given area and perimeter