What functions can I enter?

You can use standard math expressions: +, -, *, /, ^ for exponent, sqrt(), sin(), cos(), tan(), log() (base 10), ln(), exp(), abs(), and the constant PI. For example: x^3 - 2*x - 5 or cos(x) - x.

Why does the calculator say "f(a) and f(b) must have opposite signs"?

The bisection method requires a sign change across the interval to guarantee a root exists there. If f(a) and f(b) have the same sign, either there is no root or there are an even number of roots in [a, b]. Adjust your interval.

How many iterations will it run?

The calculator runs until the error falls below your chosen tolerance, up to a maximum of 100 iterations. Most practical problems converge well before that limit.

The bisection method gives an approximation. The root is guaranteed to be within ±tolerance of the reported value. Decrease tolerance for higher precision.

Can I find complex roots?

No — the bisection method only finds real roots within a given interval. For complex roots or all roots of a polynomial, a different method is needed.

Bisection Method Calculator

Enter a function f(x) and an interval [a, b] to find its root step-by-step using the bisection method.

Function & Interval

f(x) =

Use ^ for power, * for multiply, sqrt(), sin(), cos(), ln(), exp(), PI

Lower bound a

Upper bound b

Tolerance

Quick Examples

Iteration Table

n	a	b	c = (a+b)/2	f(c)	Error

Enter a function and interval, then click Find Root.

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Summary

The bisection method is a classic numerical technique for finding roots of continuous functions. Given an interval [a, b] where f(a) and f(b) have opposite signs, the method repeatedly halves the interval, converging to the root. This calculator evaluates your function expression, runs up to 100 iterations, and displays a step-by-step table showing each midpoint, function value, and error.

How it works

Enter a function of x in standard math notation (e.g. x^3 - x - 2).
Set the lower bound a and upper bound b of the search interval.
Choose a tolerance (acceptable error, e.g. 0.0001).
The calculator checks that f(a) and f(b) have opposite signs (a root must exist).
Each iteration computes the midpoint c = (a + b) / 2 and evaluates f(c).
The interval is halved based on the sign of f(c) until the error is within tolerance.

Use cases

Solve equations like x^3 - x - 2 = 0 numerically.
Verify root-finding homework with a full iteration trace.
Learn how the bisection algorithm converges step by step.
Find roots of transcendental equations such as cos(x) - x = 0.
Check convergence rate and number of iterations needed for a given tolerance.
Explore how interval width affects convergence speed.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu