Completing the Square Calculator
Turn ax² + bx + c into a(x − h)² + k with every step.
Reviewed by James Whitfield · How we check our maths
Your values
ⓘ Live results
Results
- 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.
The formula
ax² + bx + c = a(x + b/2a)² + c − b²/4a
Worked example
Worked example
Using the values the calculator loads with.
Inputs
- a3
- b12
- c7
Result
- Vertex form3(x + 2)² − 5
- Vertex(-2, -5)
- Value added inside4
- Value subtracted outside12
- Minimum or maximum value-5
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.
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.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error