1. The bisection method is a numerical technique to find roots of a continuous function $f(x)$ where the function changes sign over an interval $[a,b]$. 2. The method requires that $f(a)$ and $f(b)$ have opposite signs, i.e., $f(a) \cdot f(b) < 0$, ensuring a root exists between $a$ and $b$ by the Intermediate Value Theorem. 3. The algorithm proceeds by repeatedly bisecting the interval and selecting the subinterval where the sign change occurs. 4. At each step, compute the midpoint $c = \frac{a+b}{2}$ and evaluate $f(c)$. 5. If $f(c) = 0$ or the interval width $|b - a|$ is less than a desired tolerance, stop; $c$ is the root approximation. 6. Otherwise, determine the subinterval where the sign change occurs: - If $f(a) \cdot f(c) < 0$, set $b = c$. - Else, set $a = c$. 7. Repeat steps 4-6 until the root is approximated within the desired tolerance. This method is simple, reliable, and guarantees convergence but can be slower compared to other methods like Newton-Raphson. Final answer: The bisection method finds a root of $f(x)$ in $[a,b]$ by iteratively halving the interval where $f$ changes sign until the root is approximated within a tolerance.