Rewrite ax² + bx + c as a(x − h)² + k and expose the vertex. The page traces how the inputs become completed-square form before rounding.
The form reveals a parabola’s vertex, supports quadratic solving, and converts between expanded and geometric descriptions.
Completing the square groups a quadratic around h=−b/(2a), with k equal to the polynomial value at h. Completing the Square also connects with vertex form.
Preserve Quadratic coefficient a when checking the Completing the Square output. Keep Linear coefficient b under the same convention and estimate Completed-square form. A controlled change to Constant c should move the Completing the Square Completed-square form in a mathematically consistent direction.
Coefficient a cannot be zero. Sign handling inside x−h often causes errors when h is negative.
2x²−8x+3 becomes 2(x−2)²−5, so the vertex is (2,−5). This Completing the Square example can be compared with quadratic roots.
Compute h=−b/(2a), then k=c−b²/(4a), and verify by expanding a(x−h)²+k.
Expand the completed square after conversion. The x² coefficient, x coefficient, and constant must reproduce a, b, and c separately.
For Completing the Square, the best cross-check uses a different identity rather than entering the same formula again. Save enough context to reproduce both routes: field labels, units, angle convention, and unrounded intermediate values. Two independent paths converging on the same quantity make a copied or transposed input easier to detect.
Substitute the Completing the Square solution into Quadratic coefficient a. This Completing the Square check rejects false Completing the Square branches and forbidden denominators.
Choose an easy Quadratic coefficient a value before running Completing the Square. Predict Linear coefficient b, then compare it with the Completing the Square output.
Before using Completed-square form downstream, substitute a simple Quadratic coefficient a into the same Completing the Square relation. Preserve Linear coefficient b so the comparison remains fair. The resulting Completed-square form provides a benchmark for detecting a misplaced sign, decimal, or input order.
The vertex x-coordinate.
The quadratic value at h.
It affects both h through the denominator and the parabola’s scale.
Yes, producing a downward-opening parabola.