Completing the Square
Rewrite ax²+bx+c into vertex form a(x−h)²+k and identify the vertex coordinates.
About Completing the Square
Completing the square transforms ax²+bx+c into vertex form a(x−h)²+k, where (h,k) is the vertex of the parabola. The vertex x-coordinate is h = −b/(2a) and the y-coordinate is k = c − b²/(4a).
Formulas
Vertex coordinates
h = -b/(2a), k = c - b²/(4a)
- h
- Vertex x-coordinate
- k
- Vertex y-coordinate
- a
- x² coefficient
- b
- x coefficient
- c
- Constant
Vertex form
a(x - h)² + k
- a
- Leading coefficient
- h
- Horizontal shift
- k
- Vertical shift
Worked example
With the default values below, Completing the Square returns Vertex Form of (x - 3)² - 4. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (x² coefficient)
- 1
- b (x coefficient)
- -6
- c (constant)
- 5
Results
- Vertex Form
- (x - 3)² - 4
- h (vertex x)
- 3.000000
- k (vertex y)
- -4.000000
Step by step
- 1Values used
- 2Vertex coordinates
- 3Vertex form
- 4Vertex Form= (x - 3)² - 4
- 5h (vertex x)= 3.000000
- 6k (vertex y)= -4.000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Why is completing the square useful?
It reveals the vertex of a parabola, makes it easy to find the axis of symmetry, and is the basis for deriving the quadratic formula.
How does this relate to the quadratic formula?
The quadratic formula is derived by completing the square on the general form ax²+bx+c=0 and solving for x.