Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with complete discriminant analysis and complex root resolution.
Discriminant (b² − 4ac)
1.0000
Two distinct real roots
x₁
3
x₂
2
x = (-b ± sqrt(b²-4ac)) / 2aHow to Use the Quadratic Formula Calculator
- Enter the coefficients a, b, and c from your quadratic equation (a ≠ 0).
- The calculator computes the discriminant Δ = b² - 4ac.
- View the two roots x₁ and x₂ (real numbers or complex numbers with imaginary unit i).
Formulas & Principles
Standard Form: ax² + bx + c = 0
Discriminant: Δ = b² - 4ac
Quadratic Formula: x = (-b ± √Δ) / (2a)
Root Nature: Δ > 0 (2 Real), Δ = 0 (1 Real Repeated), Δ < 0 (2 Complex Conjugates)
Frequently Asked Questions
What does the discriminant (b² - 4ac) indicate?
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are two conjugate complex roots.
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / (2a).
What happens if coefficient a is zero?
If a = 0, the equation is no longer quadratic; it degenerates into a linear equation bx + c = 0 with a single solution x = -c/b.
Why this calculator matters
This calculator helps solve mathematical problems faster and with fewer manual arithmetic mistakes. It is useful for studying, research verification, technical engineering checks, and everyday problem-solving.
When to use it
Use it when you need a fast answer, a cross-check for manual calculations, or a clear step-by-step verification of your result.
Quick tips
Double-check your inputs, keep units consistent across all fields, and verify boundary conditions.
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