Equation Solver

• Linear (ax + b = 0) — trivial, x = -b/a.
• Quadratic (ax² + bx + c = 0) — closed form via the quadratic formula. Three discriminant cases: > 0 (two real roots), = 0 (one repeated), < 0 (complex conjugate pair).
• Cubic (ax³ + bx² + cx + d = 0) — closed form via Cardano (1545) or trigonometric form. Always has at least one real root. The formula is real-valued for three real roots but requires complex arithmetic to evaluate.
• Quartic — closed form via Ferrari (1540). Practical but tedious.
• Quintic and higher — Abel-Ruffini: no general algebraic solution exists. Numeric methods only.
• Special transcendentals — sin(x) = a, e^x = a, log(x) = a all have closed forms. Mixed (sin(x) + x = 0) generally do not.
• Linear systems Ax = b — Gauss-Jordan elimination; closed form if A is invertible.

• Polynomial roots — for degree ≤ 4 use closed-form (exact when coefficients are integers; floating point otherwise). For degree ≥ 5 or for non-polynomial equations use numeric methods (Newton-Raphson, bisection).
• Bisection — guaranteed convergence on a sign change in [a, b]. Slow (linear convergence) but bombproof. Use as fallback when faster methods fail.
• Newton-Raphson — quadratic convergence near a root, but needs a good initial guess and the derivative. Diverges in pathological cases (multiple roots, oscillating behavior).
• Brent's method — combines bisection's reliability with secant/inverse-quadratic interpolation's speed. The default in scipy.optimize.brentq. The right answer for "find a root in this interval" most of the time.
• Symbolic — if exact symbolic answers matter, use a CAS (SymPy, Maple, Mathematica) — this tool is numeric.

For Ax = b: if A is square and invertible, x = A^-1 b is the textbook answer but numerically wasteful. Production solvers use LU decomposition (general dense), Cholesky (symmetric positive-definite), or QR (rectangular least-squares). For sparse A, use specialized solvers (CG, BiCGSTAB).

If A is rectangular (more equations than unknowns), the system is overdetermined — typically no exact solution exists. Find the least-squares solution that minimizes ‖Ax - b‖². If fewer equations than unknowns (underdetermined), infinite solutions exist; pick a basis or specify a constraint (e.g., minimum-norm solution).

• You need to solve a quadratic or cubic in the middle of an interview, problem set, or design calculation and want to skip the algebra.
• You are checking the result of a manual solution against a trusted reference.
• You are computing physics or engineering quantities (projectile motion, RLC circuit analysis, beam deflection) where the equations are textbook but the algebra is tedious.
• You are debugging a numeric solver in your own code and want to compare against a known-correct result on test inputs.

• It will not do symbolic calculus or algebra. For symbolic integration, differentiation, simplification, factoring with symbolic parameters — use SymPy or Maple.
• It will not solve differential equations. Equations involving y' (derivatives) need ODE-specific tools (scipy.integrate, MATLAB ode45).
• It will not handle systems of nonlinear equations beyond simple cases. Multi-variable nonlinear systems require fsolve-style numeric solvers with good initial guesses; results depend heavily on the starting point.
• It will not validate "is this the answer you wanted". Many equations have multiple roots; the solver returns all it finds, you pick the physically meaningful one.