Completing the Square Calculator

Completing the square turns a quadratic into vertex form, revealing its turning point.

How the Math Works

Completing the square transforms a quadratic expression from standard form (ax² + bx + c) into vertex form [a(x - h)² + k]. The process uses the formulas h = -b/2a and k = c - b²/4a, where h represents the x-coordinate of the vertex and k is the y-coordinate. This transformation reveals the parabola's vertex (h, k) directly, making it easier to analyze the quadratic's maximum or minimum point, axis of symmetry, and overall shape without graphing.

Practical Applications

To use this calculation, substitute your quadratic's coefficients into the formulas. For example, with 2x² + 8x + 5, compute h = -8/(2×2) = -2 and k = 5 - 8²/(4×2) = -3, yielding vertex form 2(x - (-2))² + (-3) = 2(x + 2)² - 3. This form immediately shows the vertex at (-2, -3) and simplifies solving optimization problems, finding intercepts, or analyzing parabolic motion in physics and engineering applications.

Day-to-Day Use

This calculation helps in everyday scenarios like maximizing garden area with limited fencing, determining optimal pricing for maximum profit, or analyzing projectile motion such as a ball's trajectory. By revealing the peak value and symmetry point, completing the square provides intuitive insights for decision-making in business, sports, and design, helping you find ideal solutions quickly without complex graphing or trial-and-error methods.

Worked example

x² − 6x + 11 = (x − 3)² + 2, so the vertex is (3, 2).

FAQ