Completing the Square
Rewrite ax²+bx+c into vertex form a(x−h)²+k and identify the vertex coordinates.
Inputs
Must be non-zero
Vertex Form
(x - 3)² - 4
h (vertex x)
3.000000
k (vertex y)
-4.000000
Step by step
Values used
a (x² coefficient) = 1; b (x coefficient) = -6; c (constant) = 5
Vertex coordinates
h = -b/(2a), k = c - b²/(4a)
Vertex form
a(x - h)² + k
Vertex Form
= (x - 3)² - 4
h (vertex x)
= 3.000000
k (vertex y)
= -4.000000
How it works
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
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.