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This question requires the use of Completing the Square:

5 – 2x – x^2 = -1 ( x^2 + 2x – 5)

= -1 (x^2 + 2x + 1 – 6)

= -1 (x^2 + 2x + 1) + 6

= 6 – (x+1)^2, p =6 and q =1

(a) You can solve this graphically:

The maximum value will be at y=6

Alternatively, since (x+1)^2 has to be a non-negative value, the expression is at its maximum value when (x+1)^2 = 0. Thus, maximum value = 6

(b) 5 – 2x – x^2 = 0

Hence, 6 – (x+1)^2 = 0

(x+1)^2 = 6

(x+1) = +- sqrt 6

x = sqrt 6-1 or -sqrt 6-1//

Alternatively, use the quadratic formula and sub in a = -1, b = -2, c = 5

 

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