Completing the Square Calculator

Completing the Square Calculator — Vertex Form & Solving Quadratics

Convert quadratic expressions to vertex form and solve quadratic equations. Learn the geometric meaning and derive the quadratic formula. For related tools, see our quadratic equation solver.

Table of Contents

1. 1What Is Completing the Square?
2. 2The Geometric Meaning
3. 3The General Formula
4. 4Solving Quadratic Equations
5. 5Step-by-Step Examples
6. 6More Worked Examples
7. 7Vertex Form and Graphing
8. 8Deriving the Quadratic Formula
9. 9Common Mistakes
10. 10Real-World Applications
11. 11Practice Problems
12. 12Tips for Mastery
13. 13Final Thoughts
14. 14FAQ: Common Questions

Complete the Square

Enter coefficients a, b, c for ax² + bx + c. Click Complete the Square.

Coefficient a

Coefficient b

Coefficient c

The completed square form will appear here.

Note: Completing the square rewrites ax² + bx + c as a(x − h)² + k.

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Completing the Square – Complete Explanation

Completing the square transforms a quadratic into a perfect square trinomial plus a constant, revealing the vertex and enabling equation solving.

Completed Square Form

ax² + bx + c = a(x − h)² + k
h = −b/(2a) | k = c − b²/(4a)

1. What Is Completing the Square?

A technique to rewrite quadratics to expose vertex and roots.

2. The Geometric Meaning

Adding (b/2)² to x² + bx completes a literal square geometrically.

3. The General Formula

h = −b/(2a) | k = c − b²/(4a)

4. Solving Quadratic Equations

Isolate the squared term, take square roots, solve for x.

5. Step-by-Step Examples

x² + 8x + 12

Half of 8 = 4, square = 16
(x² + 8x + 16) − 16 + 12 = (x + 4)² − 4

2x² + 8x + 5

2(x² + 4x) + 5 → 2(x + 2)² − 3

6. More Worked Examples

x²−10x+16: (x−5)²−9

3x²+12x+7: 3(x+2)²−5

x²−3x+1: (x−3/2)²−5/4

7. Vertex Form and Graphing

Vertex form a(x−h)²+k directly gives vertex (h,k) and axis x=h.

8. Deriving the Quadratic Formula

Complete the square on ax²+bx+c=0 to derive x = [−b ± √(b²−4ac)]/(2a).

9. Common Mistakes

• Forgetting to factor out a when a≠1.
• Incorrect half calculation.
• Not subtracting what was added.
• Distribution errors.
• Sign errors with vertex.

10. Real-World Applications

• Physics: Projectile motion.
• Engineering: Optimization.
• Architecture: Parabolic arches.
• Economics: Supply/demand models.
• Computer graphics: Bezier curves.

11. Practice Problems

P1: x²+12x+20 → (x+6)²−16

P2: 4x²−8x+1 → 4(x−1)²−3

P3: Solve x²+4x−5=0 → x=1, x=−5

P4: Vertex of 2x²−8x+7 → (2,−1)

12. Tips for Mastery

• Check leading coefficient first.
• Pattern: half, square, add, subtract.
• Verify by expanding.
• Use h = −b/(2a) to check.
• Practice negatives.

13. Final Thoughts

Completing the square is a gateway to understanding quadratics. Master it thoroughly.

14. Frequently Asked Questions

A technique to rewrite ax²+bx+c as a(x−h)²+k.

Geometrically, adding (b/2)² completes a literal square.

a(x−h)²+k — reveals vertex (h,k) and axis of symmetry x=h.

Isolate the squared term, take square roots, solve for x.

The quadratic formula is derived by completing the square on ax²+bx+c=0.

There are no real solutions — the roots are complex numbers.

Expand a(x−h)²+k — you should get back ax²+bx+c.

Explore Math Topics

Continue your learning journey with our other math guides

🔢Basic Arithmetic 📐Pre-Algebra 📊Algebra 📏Geometry 📈Statistics

Related Calculators

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• 🔹 Polynomial Evaluation Calculator — Evaluate P(x).
• 🔹 Quadratic Inequality Solver — Solve quadratic inequalities.
• 🔹 Expand Binomials Calculator — Expand (x+a)² expressions.

Conclusion

Completing the square is a gateway to understanding quadratics. Master it thoroughly.

Readers who wish to continue practicing can explore the companion learning resource for guided exercises.