Quadratic Formula Calculator
Solve ax2 + bx + c = 0 using the quadratic formula.
Returns real and complex roots, vertex, discriminant, and axis of symmetry with step-by-step working.
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. It always produces a parabola when graphed and can have 0, 1, or 2 real solutions (roots).
The quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
This formula always works for any quadratic equation. The ± means there are potentially two solutions.
The discriminant (b² − 4ac) tells you how many solutions exist:
- Positive: two distinct real roots
- Zero: exactly one real root (the parabola touches the x-axis)
- Negative: no real roots (complex/imaginary roots)
Worked example: Solve 2x² − 7x + 3 = 0 Here a = 2, b = −7, c = 3
Discriminant = (−7)² − 4(2)(3) = 49 − 24 = 25 (positive → 2 solutions)
x = (7 ± √25) / (2 × 2) = (7 ± 5) / 4
x₁ = (7 + 5) / 4 = 12/4 = 3 x₂ = (7 − 5) / 4 = 2/4 = 0.5
Check: 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0 ✓
Alternative methods:
- Factoring: works when roots are nice integers or simple fractions
- Completing the square: useful for converting to vertex form
- Graphing: visual but imprecise
Real-world applications:
- Projectile motion (when does a ball hit the ground?)
- Area problems (find dimensions given area)
- Profit maximisation (price vs demand curves)
- Engineering: beam deflection, circuit resonance
- Finance: break-even analysis
Quadratics are also the foundation for understanding polynomials of higher degree and calculus optimisation.