Completing the Square Calculator — Vertex Form
Quick answer: Enter a, b, c to convert ax² + bx + c to vertex form and find the vertex.
Complete the square for any quadratic ax² + bx + c to convert it to vertex form a(x+h)² + k. Shows every step: factoring, adding and subtracting the square, and the final vertex form. Also identifies the vertex (h, k) and axis of symmetry.
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Last reviewed: April 2026Report an error
Convert ax² + bx + c to vertex form a(x + h)² + k
Vertex Form
1(x + 3)² + -4
Vertex: (-3, -4). Standard form: 1x² + 6x + 5. Vertex form: 1(x + 3)² + -4.
Vertex (h, k)
(-3, -4)
Vertex Form
1(x+3)²+-4
Step-by-step
- 1. Start: 1x² + 6x + 5
- 2. Factor out a: 1(x² + 6x) + 5
- 3. Half of 6 = 3, squared = 9
- 4. Add and subtract: 1(x² + 6x + 9) + 5 − 9
- 5. Result: 1(x + 3)² + -4
- 6. Vertex: (-3, -4)
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How to Use This Completing the Square Calculator
- 1Enter a (coefficient of x²).
- 2Enter b (coefficient of x).
- 3Enter c (constant term).
- 4Read the vertex form, vertex coordinates, and step-by-step solution.
Frequently Asked Questions
- A technique to rewrite a quadratic in vertex form a(x+h)² + k, which reveals the vertex and makes solving easier. It is also the derivation of the quadratic formula.
- a(x−h)² + k, where (h, k) is the vertex of the parabola. Positive a = opens up (minimum), negative a = opens down (maximum).
- Vertex x = −b/(2a). Vertex y = c − b²/(4a). This is the axis of symmetry and the max/min of the parabola.
- Both work for finding roots. Completing the square also reveals vertex form, which is more useful for graphing and optimization problems.
- The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. It is the algebraic proof behind the formula.
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