Exact quadratic transformation
Complete the Square Calculator
Rewrite ax² + bx + c in completed-square vertex form, show the balancing steps, and verify the result by exact re-expansion.
All entries and results stay in this browser tab. Nothing is uploaded, persisted, placed in the URL, or sent to analytics.
Enter known values
ax² + bx + c = a(x − h)² + k
Enter ordinary finite decimals. a must not be zero. Values are converted to exact fractions before transformation.
Loading local calculator…
Completed-square form
Vertex:
Expansion check:
Supplementary roots:
Method
Input decimals are normalized to reduced rational numbers. h = −b/(2a) and k = c − b²/(4a) are calculated exactly. The result is re-expanded using a, −2ah, and ah² + k; all three recovered coefficients must exactly equal the inputs.
Limitations
This owns the completing-the-square transformation. It is not a general expression parser, graphing tool, factoring engine, arbitrary-degree solver, worksheet generator, or general computer algebra system.
Privacy
No account, server calculation, remote API, fetch, XHR, beacon, cookie, local storage, session storage, IndexedDB, or user-value telemetry is used.