Completing the Square Calculator

Completing the Square Guide

The Method for a = 1

For x² + bx + c: write as (x + b/2)² − (b/2)² + c. Example: x² + 6x + 7. Half of 6 is 3. (x+3)² − 9 + 7 = (x+3)² − 2. Vertex form: (x+3)² − 2. Turning point: (−3, −2). The x-coordinate of the vertex is always −(b/2a). The y-coordinate is c − b²/(4a). Check: expand (x+3)² − 2 = x² + 6x + 9 − 2 = x² + 6x + 7 ✓.

Completing the Square for a ≠ 1

For ax² + bx + c: factor out a first. 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2(x+2)² − 8 + 3 = 2(x+2)² − 5. Turning point: (−2, −5). Alternatively: use the formula. Vertex x-coordinate: −b/(2a) = −8/(2×2) = −2. Vertex y-coordinate: substitute x = −2 into the original: 2(4) + 8(−2) + 3 = 8 − 16 + 3 = −5. The turning point is (−2, −5).

Solving Quadratics by Completing the Square

From vertex form (x+p)² = q, take square roots: x + p = ±√q. x = −p ± √q. Example: x² + 6x + 7 = 0 → (x+3)² = 2 → x = −3 ± √2 = −3 + 1.414... = −1.586 or −3 − 1.414 = −4.414. If q < 0: no real roots (parabola does not cross x-axis). If q = 0: repeated root (parabola just touches x-axis). If q > 0: two distinct roots. Completing the square derives the quadratic formula: x = (−b ± √(b²−4ac)) / 2a.

Applications of the Vertex Form

The vertex form y = a(x−h)² + k immediately shows: turning point (h, k). Whether minimum (a > 0) or maximum (a < 0). The axis of symmetry x = h. The minimum/maximum value (k). This form is essential in physics: projectile motion (height vs time is a negative quadratic — vertex gives maximum height). Economics: profit maximisation (revenue minus cost is often a downward quadratic — vertex gives the profit-maximising quantity). Engineering: cable suspension bridge profiles follow parabolic (quadra