math ToolUpdated 2026-09-20
Quadratic Formula Calculator & Solver
Enter the coefficients a, b, and c of any standard second-degree polynomial equation (ax² + bx + c = 0) to find roots, discriminant, axis of symmetry, and vertex coordinates.
Formula & Mathematical Methodology
x = (-b ± √(b² - 4ac)) ÷ (2a); Discriminant Δ = b² - 4ac
The quadratic formula provides the exact solutions to any quadratic equation. The discriminant (Δ = b² - 4ac) indicates whether roots are two distinct real numbers (Δ > 0), one repeated real root (Δ = 0), or two complex conjugate roots (Δ < 0).
Step-by-Step Worked Example
Scenario: Solve 1x² - 5x + 6 = 0
1. Identify coefficients: a = 1, b = -5, c = 6.
2. Calculate discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1.
3. Since Δ = 1 > 0, there are two distinct real roots.
4. x₁ = (-(-5) + √1) ÷ (2 × 1) = (5 + 1) ÷ 2 = 3.
5. x₂ = (-(-5) - √1) ÷ (2 × 1) = (5 - 1) ÷ 2 = 2.
Final Calculated Result: x₁ = 3, x₂ = 2
Frequently Asked Questions
What does a negative discriminant mean?
When b² - 4ac is less than zero, the square root involves a negative number, resulting in complex or imaginary roots containing the imaginary unit i (√-1).
Why cannot coefficient "a" equal zero?
If a = 0, the x² term disappears, converting the equation into a linear equation (bx + c = 0) where x = -c/b.
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