First, we square both sides to get rid of the radical.

(x – 2)^{2} = 10x + 19

x^{2} – 4x + 4 = 10x + 19

x^{2} – 14x – 15 = 0

(x – 15)(x + 1) = 0

x = 15 OR x = –1

We've done the algebra correctly and got two solutions, but we can't assume that both work. We have to check each one in the original equation.

Check x = 15:

left side = 15 – 2 = 13

right side = = 13

The two sides are equal, so x = 15 is valid solution.

Check x = –1:

left side = (–1) – 2 = –3

right side = = +3

Since the radical symbol ( √ ) means to take the positive root only, the two sides have opposite signs and so are not equal. This means that x = –1 is not a solution.

The only solution is x = 15.

Answer = (C)

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