Completing the Square Calculator
Turn ax² + bx + c into a(x − h)² + k with every step.
Enter your values
Enter the a you are working with.
Enter the b you are working with.
Enter the c you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Vertex form
- 3(x + 2)² − 5
- Vertex
- (-2, -5)
- Value added inside
- 4
- Value subtracted outside
- 12
- Minimum or maximum value
- -5
Working
- 1Factor out 3: 3(x² + 4x) + 7
- 2Half the x coefficient and square it: (2)² = 4
- 3Add and subtract inside, then simplify to the vertex form above.
Quick answer
Completing the square rewrites ax² + bx + c as a(x + b/2a)² + (c − b²/4a), exposing the vertex directly.
ax² + bx + c = a(x + b/2a)² + c − b²/4a
At a glance
| What it does | Turn ax² + bx + c into a(x − h)² + k with every step. |
|---|---|
| Category | Math |
| Inputs needed | a, b, c |
| Main output | Vertex form |
| Formula | ax² + bx + c = a(x + b/2a)² + c − b²/4a |
| Cost | Free — no sign-up, no download |
How to use the Completing the Square Calculator
- 1Enter a, b and c.
- 2Read the finished vertex form and the vertex coordinates.
- 3Follow the three working lines if you need to reproduce the steps by hand.
Inputs explained
- a
- Enter the a you are working with.
- b
- Enter the b you are working with.
- c
- Enter the c you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- a3
- b12
- c7
Results
- Vertex form3(x + 2)² − 5
- Vertex(-2, -5)
- Value added inside4
- Value subtracted outside12
- Minimum or maximum value-5
- Factor out 3: 3(x² + 4x) + 7
- Half the x coefficient and square it: (2)² = 4
- Add and subtract inside, then simplify to the vertex form above.
Frequently asked questions
Why complete the square at all?
It exposes the vertex, derives the quadratic formula, and is the standard trick for integrating and for circle and conic equations.
What if a is not 1?
Factor a out of the x terms first — the calculator shows that step explicitly.
What do I need to enter into the Completing the Square Calculator?
Just 3 values: a, b, c. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Completing the Square Calculator work out the answer?
Completing the square rewrites ax² + bx + c as a(x + b/2a)² + (c − b²/4a), exposing the vertex directly. It applies the formula ax² + bx + c = a(x + b/2a)² + c − b²/4a and shows the working so you can check each step by hand.
Is the Completing the Square Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.