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最終更新日: 2026年7月31日

Does Completing the Square Always Work?

Dharmendra Singh - Creator

レビュアー

Balram Rajput
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数式

ax² + bx + c = a(x-h)² + k where h = -b/(2a), k = c - b²/(4a)

ここで:

  • a,b,c=Quadratic coefficients
  • h,k=Vertex form parameters
Completing the SquareThe quadratic ax^2 + bx + c is transformed into vertex form a(x - h)^2 + k.ax² + bx + ca(x - h)² + kAlways works when a ≠ 0h = -b / 2ak = c - b² / 4a
Completing the square always works for a true quadratic (a ≠ 0), converting it to vertex form.

解法例

Standard quadratic

x² - 4x + 3.

  1. 1h = -(-4)/(2×1)=2
  2. 2k = 3 - (-4)²/(4×1) = -1
  3. 3a ≠ 0 so method works
最終回答: h=2, k=-1, works=Yes

Non-unit leading coefficient

2x² + 8x + 5.

  1. 1h = -8/(4) = -2
  2. 2k = 5 - 64/8 = -3
  3. 3Still valid because a ≠ 0
最終回答: h=-2, k=-3, works=Yes

Not quadratic case

0x² + 4x + 3.

  1. 1a = 0 removes quadratic term
  2. 2Expression is linear, not quadratic
  3. 3Completing-square workflow not applicable
最終回答: works=No

はじめに

This calculator checks whether completing the square applies and computes vertex-form parameters h and k when the equation is quadratic.

Formula and Method

The Does Completing the Square Always Work? uses the core relation ax² + bx + c = a(x-h)² + k where h = -b/(2a), k = c - b²/(4a).

  • Apply the formula directly to user inputs

  • Compute intermediate terms when needed

  • Round numeric outputs for stable display

  • Return structured outputs for UI rendering

Input Fields

Provide values for: a, b, c.

  1. 1

    a: a

  2. 2

    b: b

  3. 3

    c: c

Output Fields

The calculator returns: h, k, works.

  1. 1

    h: Horizontal shift in vertex form

  2. 2

    k: Vertical shift in vertex form

  3. 3

    works: Yes for quadratics (a ≠ 0), No otherwise

Validation Rules

a, b, and c must be numeric. If a = 0, the solver reports works = No.

  • Invalid or missing numeric values trigger safe fallback outputs

  • Error messaging is returned by the calculation function when constraints fail

  • Domain restrictions are enforced before formula evaluation

  • Rounding is applied to final numeric values

How to Use This Calculator

Enter valid inputs, run the calculation, and review both primary and supporting outputs.

  • Enter each required field

  • Click calculate

  • Check primary output first

  • Use examples to verify expected behavior

Practical Uses

This calculation is commonly used in math practice, engineering checks, and fast verification tasks.

  • Homework and exam preparation

  • Quick sanity checks while modeling

  • Teaching and tutoring demonstrations

  • Reusable reference for manual calculations

よくある質問

What does the Does Completing the Square Always Work? compute?

It computes h, k, works from the provided inputs.

Which inputs are required?

Required fields are a, b, c.

How are results rounded?

Numeric outputs are rounded by the calculation function (generally to about 6 decimal places) for cleaner display.

What if I enter invalid values?

a, b, and c must be numeric. If a = 0, the solver reports works = No.

Which output should I read first?

Start with works, then use the remaining outputs for context and interpretation.

Can I use decimal values?

Yes. Decimal inputs are supported unless the calculator specifically requires integer-style values.

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