Math & Conversions

Completing the Square Calculator

Rewrite a quadratic ax^2 + bx + c in vertex form by completing the square. Enter the coefficients a, b, and c and the tool returns the vertex coordinates h and k. Free, runs in your browser, no sign-up.

What this calculator does

The Completing the Square Calculator estimates your result instantly from the values you enter, using standard math & conversions formulas. Rewrite a quadratic ax^2 + bx + c in vertex form by completing the square. Enter the coefficients a, b, and c and the tool returns the vertex coordinates h and k. Free, runs in your browser, no sign-up. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.

Completing the Square Calculator

Result

—
Vertex h
—
Vertex k

About Completing the Square Calculator

The Completing the Square Calculator converts a quadratic a x² + b x + c into vertex form a(x − h)² + k. It finds the vertex using h = −b ÷ (2a) and k = c − b² ÷ (4a).

The coefficient a cannot be zero, since that would make the expression linear rather than quadratic.

Formula & Mathematical Calculation

Vertex h = f(a, b, c)

Calculated according to standard math formulas implemented in the calculator core.

Calculation Example

For a = 1, b = 6, and c = 5: h = −6 ÷ (2 × 1) = −3 and k = 5 − 6² ÷ (4 × 1) = 5 − 9 = −4. The vertex form is (x − (−3))² − 4, so the tool shows Vertex h = -3 and Vertex k = -4.

Key Benefits

  • Returns the vertex coordinates h and k directly from the coefficients.
  • Handles any quadratic where a is not zero, including negative and fractional coefficients.
  • Gives the values needed to write the vertex form a(x - h)^2 + k.
  • Free and runs in the browser with no sign-up.

How to Use this Calculator

  1. Enter coefficient a
    Type the coefficient of x squared into the a field; it cannot be zero.
  2. Enter coefficient b
    Type the coefficient of x into the b field.
  3. Enter coefficient c
    Type the constant term into the c field.
  4. Calculate
    Press Calculate to see Vertex h and Vertex k.

Assumptions & Limitations

What the calculation assumes

  • Every value you enter is treated as an exact, known amount.
  • Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
  • Results use standard published formulas and account only for the values entered above.
  • Rounding is applied for display; the underlying calculation uses your full precision.
  • Results are returned in the same units you enter, unless the tool states the output unit.

What it does not cover

  • Completing the Square Calculator produces a calculated result for reference and learning, not a certified or official measurement.
  • Real-world outcomes differ when tolerances, rounding rules or local standards apply.
  • Definitions and standards vary by region and change over time; verify the convention that applies to your case.
  • Figures are only as accurate as the inputs you supply — check them before acting on the result.
  • For work where the result carries real consequences, confirm it against an authoritative source.

Frequently Asked Questions

If a is zero the expression is linear, not quadratic, and the vertex formula divides by 2a, so the tool blocks a = 0 with an error.

They are the coordinates of the parabola's vertex, so the quadratic can be written as a(x - h)^2 + k.

The tool uses k = c - b squared divided by 4a, the constant left after completing the square.

Yes, both are rounded to four decimal places.

Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.