Quadratic Formula Calculator

    Solve quadratic equations ax² + bx + c = 0 using the quadratic formula with detailed step-by-step solutions and analysis.

    Quadratic Formula Calculator

    ax² + bx + c = 0
    1x² + -5x + 6 = 0

    About the Quadratic Formula

    The quadratic formula provides a systematic method to solve any quadratic equation ax² + bx + c = 0. It's derived by completing the square and works for all quadratic equations.

    The Formula

    x = (-b ± √(b² - 4ac)) / (2a)

    Discriminant Analysis

    • Δ > 0: Two distinct real roots
    • Δ = 0: One repeated real root (perfect square)
    • Δ < 0: Two complex conjugate roots

    Vertex Form

    • Vertex x-coordinate: x = -b/(2a)
    • Vertex y-coordinate: y = f(-b/(2a))
    • Axis of symmetry: x = -b/(2a)

    Applications

    • Physics: Projectile motion, optimization problems
    • Engineering: Structural analysis, control systems
    • Economics: Profit maximization, cost analysis
    • Geometry: Area and perimeter optimization

    Alternative Methods

    • Factoring (when possible)
    • Completing the square
    • Graphical method
    • Numerical methods for complex equations

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