Question

Complete the square to determine the minimum or maximum value of the function defined by the expression.

−x² + 10x + 5

A. maximum value at 30
B. minimum value at 30
C. maximum value at −30
D. minimum value at −30

Please provide a full explanation, thank you!

Answer 1

A. maximum value at 30

Explanation:

−x² + 10x + 5

First, factor out the leading coefficient from the first two terms:

-1 (x² − 10x) + 5

Take half of the next coefficient, square it, then add and subtract the result.

(-10/2)² = 25

-1 (x² − 10x + 25 − 25) + 5

-1 (x² − 10x + 25) + 25 + 5

-1 (x² − 10x + 25) + 30

Factor the perfect square.

-1 (x − 5)² + 30

The equation is now in vertex form. This is a downwards parabola with a vertex at (5, 30). Since the parabola points down, the vertex is a maximum.