Completing the Square Calculator
Convert any quadratic ax² + bx + c into vertex form a(x − h)² + k and instantly find its vertex and axis of symmetry.
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Horizontal shift h = −b ÷ (2a)
6 ÷ (2 × 1) = 3 - 2
b² ÷ (4a)
36 ÷ (4 × 1) = 9 - 3
Vertical shift k = c − b² ÷ (4a)
5 − 9 = -4
How does this calculator work?
To complete the square for ax² + bx + c, compute h = −b/(2a) and k = c − b²/(4a). The quadratic equals a(x − h)² + k, the vertex form. The vertex is (h, k) and the axis of symmetry is x = h, giving the parabola's turning point and minimum or maximum value.
Formula
How this is calculated
Enter the three coefficients of a quadratic written in standard form ax² + bx + c. The coefficient a must be non-zero, otherwise the expression is linear and has no parabola or vertex.
Completing the square factors a out of the first two terms and adds/subtracts the square needed to form a perfect-square trinomial. The result is the vertex form a(x − h)² + k, where h = −b / (2a) is the horizontal shift and k = c − b² / (4a) is the vertical shift. The point (h, k) is the parabola's vertex and the vertical line x = h is its axis of symmetry.
Because a is unchanged, it still controls the parabola's width and direction: a > 0 opens upward (vertex is the minimum) and a < 0 opens downward (vertex is the maximum). The k value equals the function's minimum or maximum output. All coefficients are unitless; results are exact rational expressions rounded for display.
Frequently asked questions
The vertex is (h, k) where h = −b / (2a) and k = c − b² / (4a). It is the turning point of the parabola.
If a = 0 the expression bx + c is linear, not quadratic, so it has no parabola, no square to complete, and no vertex.
When a > 0 the parabola opens upward and the vertex is a minimum; when a < 0 it opens downward and the vertex is a maximum.
Also known as
TG we-Calculate Editorial Team. (2026). Completing the Square Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/completing-the-square-calculator
TG we-Calculate Editorial Team. "Completing the Square Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/completing-the-square-calculator.
TG we-Calculate Editorial Team, "Completing the Square Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/completing-the-square-calculator
@misc{wecalculate_completing_the_square_calculator, title = {Completing the Square Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/completing-the-square-calculator}}, year = {2026}, note = {TG we-Calculate} }
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