We have the quadratic y = x^2 − x − 2 with a = 1 > 0, so the parabola opens upward and has a minimum at its vertex.
Method 1 — vertex formula:
- The x-coordinate of the vertex is x = −b/(2a) = −(−1)/(2·1) = 1/2.
- Evaluate y at x = 1/2: y = (1/2)^2 − (1/2) − 2 = 1/4 − 1/2 − 2 = −1/4 − 2 = −9/4.
Method 2 — completing the square (optional):
y = x^2 − x − 2 = (x^2 − x + 1/4) − 1/4 − 2 = (x − 1/2)^2 − 9/4.
The squared term is ≥ 0, so the minimum value is −9/4, achieved at x = 1/2.
Minimum value: −9/4.