حل المعادلة التربيعية
Solve equations of the form ax² + bx + c = 0.
الحلول
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المميِّز
Math
حول Quadratic Equation Solver
A quadratic equation (ax² + bx + c = 0) appears in projectile motion, profit maximization, geometry, and physics. Our solver uses the quadratic formula to find real and complex roots, shows the discriminant analysis, graphs the parabola, and explains the relationship between roots and the graph's x-intercepts.
كيفية الاستخدام
- Enter coefficients a, b, and c for the equation ax² + bx + c = 0.
- See both roots (real or complex), the vertex, axis of symmetry, and discriminant.
- View the step-by-step quadratic formula application.
- Completing the square method is shown as an alternative solution path.
الصيغة والمنهجية
x = (−b ± √(b²−4ac)) / 2a. Discriminant Δ = b²−4ac: Δ > 0 → two distinct real roots. Δ = 0 → one real root (repeated). Δ < 0 → two complex conjugate roots. Vertex: (−b/2a, c − b²/4a). Sum of roots: −b/a. Product of roots: c/a.
حالات الاستخدام الشائعة
- Solving projectile trajectory for maximum height and range
- Finding break-even points (revenue = cost equations often yield quadratics)
- Geometry: finding dimensions of a rectangle given area and perimeter
- Optimization: maximize area given a fixed perimeter
- Physics: gravitational free-fall time calculations
الأسئلة الشائعة
When the discriminant (b²−4ac) is negative. This means the parabola does not cross the x-axis. However, complex roots always exist: x = (−b ± i√|Δ|) / 2a. These appear in electrical engineering (AC circuit analysis) and quantum mechanics.
If a = 0, the equation reduces to a linear equation bx + c = 0 with solution x = −c/b. The quadratic formula is undefined when a = 0 (division by zero). Always check a ≠ 0 before applying the quadratic formula.
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