Quadratic Equation Solver to Find Roots Using th...

The Quadratic Formula

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It always has exactly two solutions (counting multiplicity), which may be real or complex numbers. The quadratic formula derives these solutions from the coefficients a, b, and c directly.

📊 Quadratic Equation Solver

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📐 Formula

x = (−b ± √(b² − 4ac)) / 2a

The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0: two distinct real roots; Δ = 0: one repeated real root; Δ < 0: two complex conjugate roots.

📝 Worked Example

Solve 2x² − 5x + 3 = 0:
Δ = 25 − 24 = 1
x = (5 ± √1) / 4
x₁ = (5+1)/4 = 1.5, x₂ = (5−1)/4 = 1

📝 How to Use the Calculator

Enter CoefficientsInput a, b, c from the equation ax² + bx + c = 0.

CalculateView both roots and the discriminant value.

InterpretDetermine if roots are real, repeated, or complex.

VerifySubstitute back: a(x₁)² + b(x₁) + c should equal 0.

❓ FAQ

What if the discriminant is negative?

The roots are complex numbers: x = (−b ± i√|Δ|) / 2a, where i = √(−1). Complex roots always come in conjugate pairs.

How do I factor a quadratic?

If roots are r₁ and r₂, then ax²+bx+c = a(x−r₁)(x−r₂). The quadratic formula gives you r₁ and r₂ to factor by.

The Quadratic Formula