Completing the Square Calculator

Enter ax² + bx + c to calculate h and k, show a(x − h)² + k, and connect the transformation to the vertex.

Stable numericsSubstitute / reconstructIndependent checkLocal calculation

—Vertex form a(x − h)² + k

—h

—k

—Vertex

—Square term inside

Completing the Square: Vertex, Roots & Graph Audit

Convert a quadratic to vertex form, verify the added square term, roots, discriminant, symmetry axis and a parabola graph in one audit.

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Vertex form is verified against the same coefficients

Completing the square remains a bounded ax² + bx + c transformation and is connected to h, k, roots, and the discriminant.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.

Practical guide and verification

Use the tool first, then apply these checks to verify the inputs, interpret the result, and hand it off without displacing the primary workflow.

Preserve the coefficient a through every transformation

For ax²+bx+c with a not equal to 1, factor or divide by a correctly before creating the perfect square. A common error is to halve b directly instead of the normalized x coefficient. Expand the resulting vertex form back out and confirm that the x², x, and constant coefficients match the original expression.

Separate vertex-form conversion from solving an equation

Completing the square can rewrite an expression as a(x−h)²+k even when no equation equals zero. Roots are a separate question that begins only after setting the expression equal to zero. Keeping those tasks distinct prevents a valid vertex conversion from being misread as a root calculation.

Use the discriminant as an independent root check

The sign of b²−4ac predicts whether the quadratic has two real roots, one repeated real root, or a complex-conjugate pair. Compare that classification with the roots produced after completing the square. If the root type disagrees, recheck the sign when moving the constant term or taking the square root.

Verify the vertex against symmetry and a sample point

The completed-square form gives the vertex and axis of symmetry directly. Check h against −b/(2a), then substitute a simple x value into both the original and transformed expressions. Agreement at the coefficients and at a sample point is stronger evidence than trusting a single displayed formula.