The quadratic formula calculator solves ax² + bx + c = 0 for all cases: two real roots, one repeated root, or two complex roots. Enter coefficients a, b, and c to find the roots with step-by-step solution.
Quadratic Formula Calculator (ax² + bx + c = 0)
How to Use the Quadratic Formula Calculator
The quadratic formula solves any equation of the form ax² + bx + c = 0. It always works, even when the equation can't be easily factored.
Example: x² − 5x + 6 = 0
a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25−24 = 1. x = [5 ± √1] / 2. x₁ = (5+1)/2 = 3. x₂ = (5−1)/2 = 2. Check: (x−3)(x−2) = x²−5x+6. ✓
Understanding the Discriminant
Δ = b²−4ac. Δ > 0: two distinct real roots (parabola crosses x-axis twice). Δ = 0: one repeated root (parabola touches x-axis). Δ < 0: no real roots (parabola doesn't cross x-axis); roots are complex numbers a ± bi.
Real-World Applications
Projectile height h = −½gt² + v₀t + h₀ is quadratic. To find when an object hits the ground (h=0): use the quadratic formula with a=−½g, b=v₀, c=h₀. Area problems (find dimensions from area) also lead to quadratic equations.
FAQ
What is the quadratic formula?
For ax² + bx + c = 0, the roots are x = [−b ± √(b²−4ac)] / (2a). The ± gives two solutions. If b²−4ac > 0: two real roots. If b²−4ac = 0: one repeated real root. If b²−4ac < 0: two complex conjugate roots.
What is the discriminant?
The discriminant D = b² − 4ac. D > 0: two distinct real roots (parabola crosses x-axis twice). D = 0: one real root (parabola touches x-axis). D < 0: no real roots (parabola doesn't cross x-axis), but two complex roots ±i√|D|/(2a).
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
Can you factor instead of using the quadratic formula?
For simple cases, yes. x² − 5x + 6 = (x−2)(x−3): roots are x=2 and x=3. But the quadratic formula always works. When the discriminant is not a perfect square, factoring gives irrational roots — use the formula directly. The formula is derived by completing the square.