What is the turning point when completing the square in the expression (x + m)² + n?

To determine the turning point of the expression in the form ((x + m)^2 + n), it's essential to recognize the structure of this quadratic expression. Completing the square transforms a standard quadratic equation into a vertex form, where the vertex represents the turning point on a graph.

In the expression ((x + m)^2), the term ((x + m)) indicates a horizontal shift of the parabola. Specifically, the graph of (y = (x + m)^2) shifts to the left by (m) units. The addition of (n) translates the entire graph vertically by (n) units.

The turning point, also known as the vertex of the parabola, occurs at the lowest point for a parabola that opens upwards (which is the case here, as the coefficient of the squared term is positive).

Thus, the turning point is located at the coordinates given by the horizontal shift (-m) (since (-m) results from (x + m = 0), solving for (x) gives (x = -m)) and the vertical shift (n). Therefore, this results in the turning point being at the coordinates \