What is completing the square?

Completing the square rewrites a quadratic ax² + bx + c so that the x-terms form a perfect square (x − h)², plus a leftover constant k. The trick is to take half of the x-coefficient, square it, then add and subtract that value so nothing changes but the first three terms fold neatly. For example x² + 6x + 5 becomes (x + 3)² − 4, because half of 6 is 3 and 3² = 9, leaving 5 − 9 = −4.

What is vertex form and why is a(x − h)² + k useful?

Vertex form is a(x − h)² + k. It is useful because the vertex of the parabola is read off directly as (h, k) with no extra work, and the axis of symmetry is just x = h. The k value is the minimum of the function when a > 0 and the maximum when a < 0. For x² + 6x + 5 = (x + 3)² − 4, the vertex is (−3, −4) and −4 is the minimum value.

How do I find h and k?

Use h = −b/(2a) and k = c − b²/(4a). These come straight from the algebra of completing the square. Take 2x² + 8x + 3: h = −8/(2·2) = −2 and k = 3 − 8²/(4·2) = 3 − 64/8 = 3 − 8 = −5, so the vertex form is 2(x + 2)² − 5 and the vertex is (−2, −5).

What is the axis of symmetry?

The axis of symmetry is the vertical line x = h that splits the parabola into two mirror halves, where h = −b/(2a). It always passes through the vertex. For x² − 4x + 3, h = −(−4)/(2·1) = 2, so the axis of symmetry is x = 2 and the vertex sits at (2, −1). Points equidistant left and right of x = 2 have the same y-value.

Can completing the square solve the equation, not just find the vertex?

Yes. Once you have a(x − h)² + k, set it to 0 and isolate the square: (x − h)² = −k/a, then take the square root of both sides to get x = h ± √(−k/a). If −k/a is positive you get two real roots, if it is 0 you get one repeated root, and if it is negative you get a pair of complex conjugates. For x² + 2x + 5 = (x + 1)² + 4, the solutions are x = −1 ± 2i.

When should I use completing the square instead of the quadratic formula?

Use completing the square when you need the vertex, the min/max, or the axis of symmetry, since those drop out for free. Use it to derive the quadratic formula itself, or to graph a parabola quickly. The quadratic formula is faster when you only want the numeric roots and do not care about the shape. This tool gives you both at once so you never have to choose.

Find the vertex of a parabola for graphing homework

Your assignment hands you x² − 4x + 3 and asks for the vertex and axis of symmetry before sketching. Type the coefficients, read the vertex (2, −1) and axis x = 2 straight off the vertex form (x − 2)² − 1, and you have the two anchor points the graph needs. The step list shows exactly how half of −4 became the +2 inside the square, so you can reproduce the work by hand on the test.

Solve a quadratic the way the textbook chapter requires

Some chapters insist you solve by completing the square, not the formula, and grade the method. Enter x² + 6x + 5, follow the five steps to (x + 3)² − 4, then watch the tool isolate the square and take the root to land x = −5 and x = −1. You hand in the exact method the rubric wants, with no algebra slips.

Get the minimum or maximum of a quadratic model

A profit or trajectory model like −x² + 4x − 1 has a maximum you need for the answer. The tool reports a < 0, opens downward, and the k value 3 is the maximum, reached at x = 2. No calculus, no guessing where the peak is — completing the square gives the turning point directly.

Teach the derivation with a clean worked example

Prepping a lesson on perfect-square trinomials, you want an example where every line is visible. Plug in any coefficients and the numbered steps spell out factoring a out, halving the middle term, and folding the square. Copy the vertex form into your slides and share the URL so students reopen the identical example at home.

Completing the Square Calculator — Convert ax² + bx + c to Vertex Form

Turn ax² + bx + c into vertex form a(x − h)² + k with every step shown — vertex, axis of symmetry, min/max, and solved by completing the square — browser-only

Runs locally
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Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

ax² + bx + c → a(x − h)² + k

a (quadratic)

b (linear)

c (constant)

Accepts integers, decimals (0.5), fractions (1/2), negatives (−3).

Try a textbook example

Vertex form

(x + 3)² − 4

Key features

Vertex (h, k)

(-3, -4)

Axis of symmetry

x = -3

Minimum value

-4

Opens

upward (a > 0), so k is the minimum value

Step by step

1Start from a x² + b x + c with a=1, b=6, c=5.
2The x² coefficient is already 1, so look at x² + 6x.
3Take half of the x-coefficient and square it: (3)² = 9. Add and subtract it inside.
4The first three terms fold into a perfect square: (x + 3)².
5Collect the leftover constant. The result is a(x − h)² + k = (x + 3)² − 4, with h=-3 and k=-4.

Solve by completing the square

Set (x + 3)² − 4 = 0, then isolate the square: (x − -3)² = 4.

Take the square root of both sides (it is positive), giving two real solutions:

x₁ = -5

x₂ = -1

Every step runs in your browser — your numbers never leave this tab.

What this tool does

Free completing the square calculator that rewrites any quadratic ax² + bx + c as vertex form a(x − h)² + k and shows every line of the derivation. Enter the three coefficients and the tool factors a out of the first two terms, halves the x-coefficient and squares it, folds the perfect-square trinomial into (x − h)², and collects the leftover constant into k. It reports the vertex (h, k), the axis of symmetry x = h, whether the parabola opens up or down, and the resulting minimum or maximum value. From the vertex form it then solves the equation by completing the square rather than the quadratic formula, handling two real roots, a repeated root, and complex conjugate roots. One click copies the vertex form, and the a/b/c values live in the URL so a shared link reopens the exact same worked example. Everything runs in your browser with no upload.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Completing the Square Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Find the vertex of a parabola for graphing homework
Your assignment hands you x² − 4x + 3 and asks for the vertex and axis of symmetry before sketching. Type the coefficients, read the vertex (2, −1) and axis x = 2 straight off the vertex form (x − 2)² − 1, and you have the two anchor points the graph needs. The step list shows exactly how half of −4 became the +2 inside the square, so you can reproduce the work by hand on the test.
Solve a quadratic the way the textbook chapter requires
Some chapters insist you solve by completing the square, not the formula, and grade the method. Enter x² + 6x + 5, follow the five steps to (x + 3)² − 4, then watch the tool isolate the square and take the root to land x = −5 and x = −1. You hand in the exact method the rubric wants, with no algebra slips.
Get the minimum or maximum of a quadratic model
A profit or trajectory model like −x² + 4x − 1 has a maximum you need for the answer. The tool reports a < 0, opens downward, and the k value 3 is the maximum, reached at x = 2. No calculus, no guessing where the peak is — completing the square gives the turning point directly.
Teach the derivation with a clean worked example
Prepping a lesson on perfect-square trinomials, you want an example where every line is visible. Plug in any coefficients and the numbered steps spell out factoring a out, halving the middle term, and folding the square. Copy the vertex form into your slides and share the URL so students reopen the identical example at home.

Common pitfalls

Forgetting to factor a out first when a ≠ 1. For 2x² + 8x + 3 you must pull the 2 out of the x-terms before halving, otherwise you halve 8 instead of 4 and the square comes out wrong. The vertex form is 2(x + 2)² − 5, not 2(x + 4)² + something.
Adding the squared term without subtracting it back. Completing the square adds (b/2)² inside, so you must subtract the same amount to keep the expression equal. Skipping the subtraction silently changes the constant and shifts k.
Getting the sign of h backwards. In a(x − h)² + k the vertex is at +h, so (x + 3)² puts the vertex at x = −3, not +3. The bracket sign and the vertex sign are always opposite.

Privacy

Every step — factoring, halving, folding the square, the vertex, the min/max and the solved roots — is plain JavaScript running in your browser tab. The coefficients you type never leave the page and nothing is logged. The one caveat: a, b and c are stored in the URL query string so a shared link reopens the same example, which means a "share link" pasted into chat records those three numbers in the recipient server's access log. For private work, use the copy button and paste the vertex form text instead of sharing the URL.

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Completing the Square Calculator — Convert ax² + bx + c to Vertex Form

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Completing the Square Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Find the vertex of a parabola for graphing homework

Solve a quadratic the way the textbook chapter requires

Get the minimum or maximum of a quadratic model

Teach the derivation with a clean worked example

Common pitfalls

Privacy

FAQ

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