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Algebra

Completing the Square Calculator

Complete the square for quadratics. Free online Completing the Square Calculator for algebra — instant, accurate results, mobile-friendly, no signup needed.

For ax² + bx + c = 0, show completing the square form.

Vertex (h, k)
(3.0000, -4.0000)
Form a(x−h)² + k
1(x − 3.0000)² + -4.0000

Derivation

  1. ├── 01Givena = 1, b = -6, c = 5
  2. ├── 02FormulaVertex (h, k): {let t=e.a,a=e.b,n=e.c;return"(".concat((-a / (2 × t)).toFixed(4),", ").concat((n-a² / (4 × t)).toFixed(4),")")}
  3. ├── 03Substitute{let t=e.1,1=e.-6,n=e.5;return"(".concat((-1 / (2 × t)).toFixed(4),", ").concat((n-1² / (4 × t)).toFixed(4),")")}
  4. ├── 04Compute Vertex (h, k)—
  5. ├── 05FormulaForm a(x−h)² + k: {let t=e.a,a=e.b,n=e.c;return"".concat(t,"(x \u2212 ").concat((-a / (2 × t)).toFixed(4),")\xb2 + ").concat((n-a² / (4 × t)).toFixed(4))}
  6. ├── 06Substitute{let t=e.1,1=e.-6,n=e.5;return"".concat(t,"(x \u2212 ").concat((-1 / (2 × t)).toFixed(4),")\xb2 + ").concat((n-1² / (4 × t)).toFixed(4))}
  7. └── 07Compute Form a(x−h)² + k—
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§01What is

Understanding the Completing the Square Calculator

The Completing the Square Calculator computes Vertex (h, k) from 3 inputs: a, b, c. Complete the square for quadratics.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Completing the Square Calculator sits in that toolkit — it complete the square for quadratics. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Vertex (h, k) = {let t=e.a,a=e.b,n=e.c;return"(".concat((-a / (2 × t)).toFixed(4),", ").concat((n-a² / (4 × t)).toFixed(4),")")} | Form a(x−h)² + k = {let t=e.a,a=e.b,n=e.c;return"".concat(t,"(x \u2212 ").concat((-a / (2 × t)).toFixed(4),")\xb2 + ").concat((n-a² / (4 × t)).toFixed(4))}

Where

a
a
b
b
c
c
Vertex (h, k)
Output value
Form a(x−h)² + k
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: a = 1, b = -6, c = 5.

  1. 01Start by noting the input — a: 1.
  2. 02Start by noting the input — b: -6.
  3. 03Start by noting the input — c: 5.
  4. 04Substitute these values into the formula: Vertex (h, k) = {let t=e.a,a=e.b,n=e.c;return"(".concat((-a / (2 × t)).toFixed(4),", ").concat((n-a² / (4 × t)).toFixed(4),")")} | For…
  5. 05Compute Vertex (h, k): the calculator returns (3.0000, -4.0000).
  6. 06Compute Form a(x−h)² + k: the calculator returns 1(x − 3.0000)² + -4.0000.
  7. 07Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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