Trap a root between two points where the curve has opposite signs — it MUST cross somewhere between. Check the midpoint, keep the half that still straddles zero, repeat. The bracket halves every step, so the root can’t escape. It’s slower than [[the-newton-method|Newton]] but it comes with a guarantee: give it a valid bracket and it always converges. Slide to halve the interval down to the root.
Bisection brackets a root in [a,b] where f(a) and f(b) have opposite signs; by the Intermediate Value Theorem a continuous f must cross zero inside. Evaluate the midpoint m, replace whichever endpoint keeps the sign change, and the interval HALVES each step — the error is bounded by (b−a)/2ⁿ after n steps (linear convergence, one bit per step, ~3.3 steps per decimal digit). It is slow but unconditionally convergent given a valid bracket, making it the safe fallback (and the guard inside hybrid solvers like Brent’s). A fail-loud self-check throws unless it brackets the root of x³−x−2 to 1e−9. ◆ real numerical methods, node-verified.
Bisection needs a sign-change bracket to start (it can’t find even-multiplicity roots or complex roots) and converges only linearly — far slower than Newton. The halving bound and guaranteed convergence are exact.