Quadratic Formula Calculator

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.