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Advanced polynomial equation solver for linear, quadratic, cubic, and quartic equations. Multiple solution methods including quadratic formula, factoring, and completing the square.
Equation Preview
1x² + -5x + 6 = 0
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Algebra & Polynomial Math
Solve ax² + bx + c = 0 by quadratic formula, factoring, or completing the square. Interpret real and complex roots, the discriminant, parabola geometry, and Vieta's formulas.
Quadratic Formula
x = (−b ± √D) / 2a
Discriminant
D = b² − 4ac
Sum of Roots
r₁ + r₂ = −b/a
Product of Roots
r₁ × r₂ = c/a
✓ Reviewed by the CalculatorApp Mathematics & Algebra Team
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. Its graph is a parabola: opening upward if a > 0, downward if a < 0. The solutions (roots) are where the parabola crosses the x-axis — real if D ≥ 0, complex (conjugate pair) if D < 0.
Quadratic equations model projectile trajectories, profit maximization, circuit resonance, structural deflection, and lens optics. The quadratic formula — derived by completing the square — always yields both roots regardless of the discriminant sign.
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2aWorks for all quadratics
Factoring
(x − r₁)(x − r₂) = 0Fastest when roots are integers
Completing the Square
(x + b/2a)² = (b²−4ac)/4a²Derives the formula itself
Vieta's Formulas
r₁+r₂=−b/a, r₁r₂=c/aRelate roots to coefficients
| Discriminant D | Root Type | Graph Behavior | Action |
|---|---|---|---|
| D > 0 | 2 distinct real roots | Parabola crosses x-axis twice | Compute both: x = (−b ± √D) / 2a |
| D = 0 | 1 repeated real root | Parabola touches x-axis at vertex | x = −b / 2a (single root) |
| D < 0 | 2 complex conjugate roots | Parabola does not cross x-axis | x = (−b ± i√|D|) / 2a |
| a = 0 | Not quadratic — linear equation | Straight line (bx + c = 0) | x = −c / b if b ≠ 0 |
~2000 BC — Babylonian
Solved quadratic-type area problems numerically, without symbolic algebra.
~600 BC — Indian Vedic
Brahmagupta describes rules for solving quadratics including negative solutions.
830 AD — Al-Khwarizmi
Formalizes algebraic solution methods in Al-Kitab al-mukhtasar; the word algebra derives from this work.
16th century — European
Cardano and Ferrari extend to cubics and quartics; Vieta introduces symbolic notation.
17th century — Descartes
Introduces the Cartesian plane — quadratics become parabolas visualized geometrically.
Modern era — Applied
Quadratics model projectile motion, profit optimization, electrical resonance, and optics.
Deep mathematical treatment of quadratic equations, discriminants, and root classification.
External linkFree lessons on factoring, completing the square, and the quadratic formula.
External linkAuthoritative mathematical functions and polynomial reference data.
External linkComprehensive algebra and calculus notes widely used in universities.
External linkFree lecture notes and problem sets covering polynomials and complex numbers.
External linkVisual explanations of algebraic concepts including parabolas and roots.
❌ Myth: A quadratic equation always has two different real roots.
✅ Fact: When D = 0 there is exactly one repeated root; when D < 0, both roots are complex (not real).
❌ Myth: You cannot take the square root of a negative number.
✅ Fact: You can — using imaginary numbers. √(−4) = 2i, where i = √(−1). The roots are complex conjugates: a ± bi.
❌ Myth: Factoring is always possible for any quadratic.
✅ Fact: Factoring over the integers is only possible when the discriminant is a perfect square. Otherwise, use the quadratic formula.
❌ Myth: The coefficients a, b, c must all be integers.
✅ Fact: The quadratic formula works for any real (or complex) coefficients. Decimal or fractional coefficients are fine.
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