Solve equations of various types: linear, quadratic, cubic, systems of linear equations, numeric root finding, and inequalities.
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Newton-Raphson Method: xn+1 = xn - f(xn) / f'(xn). Iteratively approximates a root of f(x) = 0.
The derivative is computed symbolically.
Requires a good initial guess x₀ near the desired root.
Inequality Solver: Solve polynomial inequalities. Supports <, <=, >, >=.
Uses a symbolic solver. The result shows the solution intervals.
Example: x^2 - 5x + 6 < 0 gives solution 2 < x < 3.
Quadratic: Solve ax² + bx + c = 0. Shows discriminant, root formula, and Vieta's formulas verification.
Cubic: Solve ax³ + bx² + cx + d = 0. Shows all real and complex roots.
Linear: Solve ax + b = 0. System 2x2/3x3: Cramer's rule for systems of linear equations.
Newton-Raphson: Numeric root finding. Inequality: Polynomial inequality solver via symbolic computation.
Analytic vs numeric solutions
An analytic (closed-form) solution expresses the answer as an exact formula — the quadratic formula x = (−b ± √(b²−4ac)) / 2a solves every quadratic in one shot. A numeric solution instead starts from a guess and iterates closer and closer, as Newton-Raphson does; it works on almost any equation but returns an approximation to a chosen precision.
General radical formulas exist for degrees 1–4 and stop there — the Abel-Ruffini theorem proves no such formula can exist for all quintics, which is exactly why numeric solvers are not a lazy shortcut but a mathematical necessity.
