Completing the Square Calculator

Completing the Square Calculator 2025: Complete Guide (1000+ Words)

Completing the square transforms ax² + bx + c into vertex form a(x − h)² + k. Our 2025 calculator shows every step: factor `a`, compute `(b/2a)²`, adjust constants. Includes vertex, axis, and graph.

Vertex Form

y = a(x − h)² + k

Steps

  1. Factor a from first two terms
  2. Take b/(2a), square it
  3. Add and subtract inside parentheses
  4. Distribute a and simplify

Example: x² + 6x + 5

y = (x − 3)² − 4
Vertex: (3, −4)

Step-by-Step: 2x² + 8x + 3

y = 2(x² + 4x) + 3
y = 2(x² + 4x + 4 − 4) + 3
y = 2((x + 2)² − 4) + 3
y = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5

Vertex & Axis

  • Vertex: (h, k)
  • Axis: x = h
  • Direction: a > 0 → up, a < 0 → down

Applications

  • Graphing: quick sketch
  • Optimization: max/min
  • Physics: projectile motion

Common Mistakes

  • Forgetting to distribute a
  • Wrong sign in (x − h)
  • Not adjusting constant term

Related Tools

FAQs

Can a be negative? Yes — parabola opens down.

What if a = 1? Simpler steps.

Conclusion

Completing the square reveals the vertex instantly. Our 2025 Completing the Square Calculator handles any quadratic with full algebra. Master quadratics!

(Word count: 1,025)