Quadratic Equation Solver
Free quadratic equation solver. Enter a, b, c for ax² + bx + c = 0 and get roots using the quadratic formula with step-by-step working.
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Quadratic Solver: ax² + bx + c = 0
Comprehensive Guide & Practice
Master the mathematical principles, formulas, and step-by-step methods with worked examples and FAQs.
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1. The Quadratic Formula
A quadratic equation takes the form ax² + bx + c = 0, where a ≠ 0. The quadratic formula gives the solutions (roots):
x = (−b ± √(b² − 4ac)) / 2a
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2. The Discriminant
The discriminant is the expression under the square root: Δ = b² − 4ac. It tells you the nature of the roots before solving:
- Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
- Δ = 0: One repeated real root (parabola touches x-axis)
- Δ < 0: Two complex roots (parabola doesn't cross x-axis)
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3. Example: Solve x² − 5x + 6 = 0
a=1, b=−5, c=6. Δ = (−5)² − 4(1)(6) = 25 − 24 = 1. x = (5 ± 1) / 2. x₁ = 3, x₂ = 2. Verify: (x−3)(x−2) = x² − 5x + 6 ✓
See also: Graphing Calculator | Equation Solver | Quadratic Formula Guide