Completing the Square Calculator

Convert ax² + bx + c to vertex form a(x - h)² + k by completing the square. Find the vertex, axis of symmetry, and whether the parabola opens up or down.

∑ Algebra📐 a(x - h)² + k where h = -b/(2a)🔢 Math
Coefficient a (x² term)
Coefficient b (x term)
Constant c
Please enter valid values.

Formula & Reference

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Completing the Square Calculator—a(x - h)² + k where h = -b/(2a)vertex (h, k)

Step-by-Step Examples

Example 1
x² − 6x + 5

Complete the square.

  • h = −(−6)/(2×1) = 3
  • k = 5 − (36/4) = 5 − 9 = −4
  • Vertex form: (x−3)² − 4
  • Vertex at (3, −4), opens up
✓ (x−3)² − 4, vertex (3, −4)
Example 2
2x² + 8x + 1

a=2, b=8, c=1.

  • h = −8/(2×2) = −2
  • k = 1 − 64/8 = 1 − 8 = −7
  • Vertex form: 2(x+2)² − 7
✓ 2(x+2)² − 7
Example 3
−x² + 4x − 3

Find maximum.

  • h = −4/(2×(−1)) = 2
  • k = −3 − 16/(−4) = −3 + 4 = 1
  • Vertex (2, 1) is a maximum
✓ Maximum at (2, 1)

Real-World Applications

📚
Algebra
Essential technique for deriving the quadratic formula and finding vertex form.
📊
Optimization
Find maximum profit or minimum cost when modeled by a quadratic.
🏀
Physics
Projectile motion: find maximum height from quadratic trajectory.
📊
Graphing
Vertex form makes graphing parabolas much easier than standard form.

Common Mistakes to Avoid

⚠️
Forgetting to factor out a first

If a ≠ 1: factor a from the x² and x terms before completing the square. Then remember to multiply the added term by a.

⚠️
Sign error on h

h = −b/(2a). The vertex is at x = h, meaning the form is (x − h)². If h = 3, write (x − 3)², not (x + 3)².

⚠️
Not adjusting k correctly

When you add (b/2a)² inside, you must subtract it outside (or subtract a×(b/2a)² if a ≠ 1) to keep the equation balanced.

Frequently Asked Questions

Why complete the square? ▾
Converts standard form ax²+bx+c to vertex form a(x−h)²+k, revealing the vertex directly. Also used to derive the quadratic formula and solve non-factorable quadratics.
What is the vertex? ▾
The turning point of the parabola. Minimum if a > 0 (opens up), maximum if a < 0 (opens down). Located at (h, k) in vertex form.
How does this relate to the quadratic formula? ▾
The quadratic formula is derived by completing the square on ax²+bx+c=0. Solving gives x = (−b ± √(b²−4ac))/(2a).
What is the axis of symmetry? ▾
The vertical line x = h passing through the vertex. The parabola is symmetric about this line.
When should I use completing the square vs the formula? ▾
The quadratic formula always works. Completing the square is useful when you need vertex form for graphing or optimization problems.

Related Math Calculators

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Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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