Root Finder
A root of f is an x-value where f(x) = 0 — the points where the graph crosses or touches the x-axis. Enter a function and a search interval below to find every root inside it.
Find roots
How the calculation works
The tool first scans the interval for sign changes, then refines each bracketed root with Brent’s method — a robust algorithm that combines the safety of bisection with the speed of secant and inverse-quadratic interpolation. It is the same routine the graphing calculator uses for its root analysis.
Roots where the function merely touches the axis without changing sign (like x² at x = 0) are found by a separate extrema-aware scan, since pure sign-change detection would miss them. Every reported root is verified by evaluating f at the result.
Tips for good results
Choose an interval that brackets the roots you care about: the tool only searches where you tell it to. For x² − 4 on [−10, 10] it finds −2 and 2; shrink the interval to [0, 10] and it reports just 2.
If no roots are reported, either the function truly has none on the interval (like x² + 1 on the real line) or the roots sit exactly at your interval endpoints — nudge the bounds slightly and try again.
Frequently asked questions
Can it find complex (non-real) roots?
No — this tool finds real roots only. Functions like x² + 1 have no real roots, so the tool honestly reports none on any real interval.
Why did it miss a root I can see on the graph?
The most common cause is a root exactly at an interval endpoint or a root the scan step jumps over in a wildly oscillating function. Narrow the interval around the suspected root and search again.
How accurate are the reported roots?
Brent’s method converges to near machine precision; reported values are rounded to 6 decimal places. Plugging a reported root back into f(x) gives a value extremely close to zero.