Quadratic Equation Solver
Solve second-degree polynomial equations ax² + bx + c = 0 using the quadratic formula. Provides exact real or complex roots, discriminant categorization, and parabolic vertex extrema.
Quadratic coefficient (must be non-zero for true parabola)
Linear coefficient
Constant y-intercept term
Values of x satisfying ax² + bx + c = 0
How to Use This Calculator
- Enter coefficient a (for x²).
- Enter coefficient b (for x).
- Enter constant c.
- View the roots, discriminant, and parabola vertex instantly.
Formula & Calculation Methodology
The quadratic formula derives from completing the square. The discriminant Δ = b² − 4ac determines the character and number of solutions.
Worked Example
Solving x² − 5x + 6 = 0
Solving x² + 4 = 0 (complex roots)
Frequently Asked Questions
What occurs if the discriminant is negative?↓
The square root of a negative number yields imaginary numbers. The equation has two complex conjugate roots (a ± bi) and never crosses the x-axis.
What if coefficient a is zero?↓
If a = 0, the equation degrades into a linear equation bx + c = 0 with a single solution x = −c/b.
Does the vertex represent a minimum or maximum?↓
If a > 0, the parabola opens upward, making the vertex a minimum. If a < 0, it opens downward, making the vertex a maximum.
What is a limitation of floating quadratic solvers?↓
When b² is much larger than 4ac, catastrophic floating cancellation can occur in standard subtraction; robust quadratic implementations preserve numerical stability.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.