Question

Complete the square to determine the minimum or maximum value of the function defined by the expression. −x2 − 14x + 1

Answer 1

• maximum: 50

Explanation:

The negative coefficient of x^2 tells you the parabola opens downward. (Any even-degree polynomial with a negative leading coefficient will open downward.)

Going through the steps for completing the square, we ...

1. Factor out the leading coefficient from the x-terms

-1(x^2 +14x) +1

2. Add the square of half the x-coefficient inside parentheses, subtract the same amount outside parentheses.

-1(x^2 +14x +49) -(-1·49) +1

3. Simplify, expressing the content of parentheses as a square.

-(x +7)^2 +50

4. Compare to the vertex form to find the vertex. For vertex (h, k), the form is

a(x -h)^2 +k

so your vertex is ...

(h, k) = (-7, 50) . . . . . . . . . a = -1 < 0, so the curve opens downward. The vertex is a maximum.

The maximum value of the expression is 50.