Solve essential algebraic equations, numerical calculus, and mathematical limits in your browser. Handles quadratic equations ax² + bx + c = 0 (with real and complex conjugate roots using the numerically stable Citardauq formula), cubic equations via Cardano formula, and 2×2 linear equation systems.
Also features a numerical calculus engine: evaluates first-order derivatives f'(x₀) via a 5-point symmetric stencil, definite integrals ∫[a,b] f(x) dx via Adaptive Simpson Quadrature, and limits (including indeterminate forms like sin(x)/x as x→0) via Richardson extrapolation across geometric perturbation sequences.
Frequently asked questions
Can it solve quadratic equations with negative discriminant (Δ < 0)?
Yes. When Δ < 0, it computes both complex conjugate roots in the standard format u ± vi.
How does it solve indeterminate limits like sin(x)/x as x → 0?
Direct substitution gives 0/0 (NaN). The solver samples decreasing perturbations h = 10⁻¹ ... 10⁻⁶ around x₀ and applies Richardson extrapolation (10·f(h/10) - f(h))/9 to eliminate first-order truncation error and cleanly converge to 1.
How is the definite integral computed?
It uses Adaptive Simpson Quadrature with recursive sub-interval refinement (tolerance 1e-9), adjusting sample density where curvature is high while computing smooth intervals in fractions of a millisecond.
中文说明
在浏览器本地实时求解经典代数方程、数值微积分与数学极限：支持一元二次方程（采用高精度 Citardauq 公式消除巨额灾难性抵消，自动输出实根与共轭复数虚根）、基于卡尔丹公式的一元三次方程解析求解，以及二元一次线性方程组。
同时集成高精度数值微积分与极限求解器：利用五点对称差商计算导数切线斜率 f'(x₀)，利用自适应辛普森积分法（Adaptive Simpson Quadrature）以极少函数采样高精度求解定积分，并通过几何步长理查森外推算法（Richardson Extrapolation）稳健化解 0/0 不定型并精确计算单侧与双侧极限（如 sin(x)/x 在 x 趋向 0 时的极限为 1）。
常见问题
判别式 Δ < 0 时能求解虚根吗？
可以。当 Δ < 0 时，计算器会自动输出共轭复数解 u ± vi。
如何求解像 sin(x)/x 在 x 趋向 0 这样的 0/0 不定式极限？
直接代入会产生 0/0 无法求值。求解器在趋近点附近按几何递减步长 h = 10⁻¹ ... 10⁻⁶ 采样，并通过理查森外推公式 (10·f(h/10) - f(h))/9 消除主导截断误差，稳定消除浮点抖动并精确收敛至极限值 1。
定积分是如何计算的？
基于自适应辛普森积分法（Adaptive Simpson Quadrature），递归评估局部误差（容差 1e-9），在平滑区域只需十余次计算即可完成，在剧烈振荡区域自动加密网格，兼顾极高精度与毫秒级速度。