Quadratic Formula Calculator

About the Quadratic Formula Calculator

The quadratic formula solves any equation in the form ax² + bx + c = 0, finding the x-intercepts (roots) of a parabola. It is one of the most important formulas in algebra and appears throughout physics (projectile motion), engineering (signal processing), and economics (optimisation). Our calculator provides step-by-step working, including the discriminant, and handles real, equal, and complex roots.

How It Works

x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac determines the nature of roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, two complex conjugate roots. For x² − 5x + 6 = 0 (a=1, b=−5, c=6): Δ = 25 − 24 = 1. x = (5 ± 1) / 2. So x = 3 or x = 2.

Tips & Best Practices

• ✓Always check: if b² − 4ac is negative, roots are complex (no real x-intercepts — parabola doesn't cross x-axis).
• ✓Verify answers by substituting back: a(root)² + b(root) + c should equal 0.
• ✓For simple quadratics, factoring is faster: x² − 5x + 6 = (x−2)(x−3).
• ✓Completing the square is an alternative method that also derives the vertex form.
• ✓Projectile motion: h(t) = −16t² + v₀t + h₀ is a quadratic in time t.

Who Uses This Calculator

Algebra students solving homework, engineers finding system equilibrium points, physicists calculating projectile landing positions, economists finding market equilibrium prices, and programmers implementing quadratic collision detection all use the quadratic formula.