Equations as constraints: linear, quadratic, and everything between
A linear equation in one variable — 3x + 5 = 20 — states that two expressions name the same number. Solving it applies the same operation to both sides until the variable stands alone: subtract 5, divide by 3, and x = 5. Each step preserves equality because it is reversible. That reversibility is the entire justification for algebra; no step may destroy information, or you risk inventing or losing solutions.
A quadratic equation, ax² + bx + c = 0, is the simplest nonlinear constraint, and its solution formula,
x = (-b ± sqrt(b² - 4ac)) / (2a),
follows by completing the square — literally rearranging the expression into a perfect square plus a remainder. The expression under the root, b² - 4ac, is the discriminant, and it earns its name: a positive discriminant gives two distinct real roots, zero gives one repeated root, and a negative gives two complex conjugates. Reading the discriminant first tells you what kind of answer to expect before you compute anything.
Higher-degree polynomials follow the same logic with more branches. The fundamental theorem of algebra guarantees exactly as many complex roots as the degree, counting multiplicity. Cubics and quartics have closed-form solutions, but from degree five upward no general formula exists (the Abel–Ruffini theorem) — a fact that pushed mathematics toward numerical methods rather than deeper formulas.