First let us note that these square roots only produce real values when the quadratic expressions are non-negative.

√(5 + 4x – x²) = √{(5 - x)(1 + x)}, so only produces real values when -1 < x < 5.

√(8 + 2x – x²) = √{(4 - x)(2 + x)}, so only produces real values when -2 < x < 4.

Therefore we must consider only what happens when -1 < x < 4.

The first expression takes its maximum value of 3 at x = 2, as 5 + 4x – x²= 9 - (x - 2)² .

The second expression takes its maximum value of 3 at x = 1, as 8 + 2x – x²= 9 - (x - 1)² .

The graph y = √(5 + 4x – x²) is obtained by translating the graph y = √(8 + 2x – x²) one unit horizontally in the positive x direction.

Each graph is also symmetrical about the vertical line through its maximum point, so the maximum value of √(5 + 4x – x²) + √(8 + 2x – x²) must occur at the midpoint of x = 1 and x = 2, ie at x = 1.5.

Substituting x = 1.5 into √(5 + 4x – x²) + √(8 + 2x – x²), the maximum value reached is √35, so for the second question we need n = 35,

The minimum value must be reached when one of the square roots is equal to zero, for example at x = -1.

Substituting x = -1 into √(5 + 4x – x²) + √(8 + 2x – x²), the minimum value reached is √5 as stated.

Challenge 351:Square Root Sum