What is the set of all solutions to the equation `sqrt(x+2)=-x` ?

Options

A. {−1, 2}
B. {−1}
C. {2}
D. There are no solutions to the given equation.

Answer

B

Explanation

Squaring both sides of the given equation yields `x + 2 = x^2`. Subtracting x and 2 from both sides of `x + 2 = x^2` yields `x^2 − x − 2 = 0`. Factoring the left-hand side of this equation yields (x − 2)(x + 1) = 0. Applying the zero product property, the solutions to (x − 2)(x + 1) = 0 are x − 2 = 0, or x = 2 and x + 1 = 0, or x = −1. Substituting x = 2 in the given equation gives `sqrt4` = −2 , which is false because `sqrt4` = 2 by the definition of a principal square root. So, x = 2 isn’t a solution. Substituting x = −1 into the given equation gives `sqrt1` = −(−1) , which is true because −(−1) = 1. So x = −1 is the only solution.
Choices A and C are incorrect. The square root symbol represents the principal, or nonnegative, square root. Therefore, in the equation `sqrt(x+2)=-x`, the value of -x must be zero or positive . If x=2, then -x=-2 , which is negative, so 2 can’t be in the set of solutions. Choice D is incorrect and may result from incorrectly reasoning that −x always has a negative value and therefore can’t be equal to a value of a principal square root, which cannot be negative.