The minimum value of (x - 2)(x - 9) is

For a quadratic expression ax2 + bx + c with a > 0, the graph is an upward-opening parabola, so the expression reaches its least value at the vertex x = −b/(2a), and that least value is simply the expression evaluated at this vertex.

Applying this to (x − 2)(x − 9):

1. Expand the product: (x − 2)(x − 9) = x2 − 11x + 18, so a = 1, b = −11, c = 18.

2. Locate the vertex: x = −b/(2a) = 11/2.

3. Substitute x = 11/2 into the factored form: (11/2 − 2)(11/2 − 9) = (7/2)(−7/2) = −49/4.

Cross-check using the roots directly: since (x − 2)(x − 9) = 0 at x = 2 and x = 9, the vertex of the parabola lies exactly midway between the roots, at x = (2 + 9)/2 = 11/2 — the same point used above, which confirms the result.

So the minimum value of (x − 2)(x − 9) is −49/4.