What Is Completing the Square?

Completing the square is an algebraic technique that rewrites a quadratic expression ax² + bx + c in the form a(x − h)² + k. The new form reveals the parabola's vertex (h, k) directly — the turning point where the graph reaches its minimum or maximum. It is the standard method for solving quadratics without the quadratic formula and for converting circles and ellipses into center-radius form.

How the Calculator Works

Enter the coefficients a, b, and c of your quadratic. The calculator computes h = −b/(2a) and k = c − b²/(4a), builds the vertex form, and reports the vertex, axis of symmetry, discriminant, and roots. With x² + 6x + 5, the result is (x + 3)² − 4, meaning the parabola's vertex sits at (−3, −4) and it crosses the x-axis at x = −1 and x = −5.

Why Vertex Form Matters

The vertex is the most informative point on a parabola. For a projectile, it is the maximum height; for a profit function, the profit-maximizing output; for an arch, the peak of the span. Because a(x − h)² + k shows the vertex explicitly, it turns a three-term expression into a description of shape: h shifts the graph horizontally, k vertically, and a controls how steeply it opens.

The Discriminant and Roots

The discriminant b² − 4ac tells you how the parabola meets the x-axis before you ever solve for roots. Positive: two real roots (the graph crosses twice). Zero: one repeated root (the vertex touches the axis). Negative: no real roots (the graph never crosses). This calculator reports all of it in one view, with a plot of the parabola and its vertex marked for visual confirmation.