Verified answer

you first note that the x value has to be negative

then square both sides and solve for x ...{ - 2 }

Be careful not to fall prey to believing in superfluous solutions.

√(x + 6) = - x ==> x + 6 = x² ==> x² - x - 6 = 0 ==> (x - 3)(x + 2) = 0.

The quadratic has two roots 3, and -2. However, the original equation has only one solution x = -2.

To see this note that if x = -2, then

√(x + 6) = √(-2 + 6) = √(4) = 2 = - (-2) = -x.

But if x = 3, then

√(x + 6) = √(3 + 6) = √(9) = 3 ≠ -3 = -x.

The symbol "√" denotes only the principal (non-negative) square root!

If it is unclear to you why there is only one real solution, you might want to ask your teacher to clarify.

sqrt (x + 6) = -x

Square both sides

(x + 6) = x^2

x^2 - x - 6 = 0

(x - 3)(x + 2) = 0

x = 3

x = - 2

Proof

sqrt(3 +6) = -3

sqrt(9) = 3

sqrt(-2 + 6) = -2

sqrt(4) = -2

There are only 2 distinct real solutions, 3 & -2