What is the value of x? (1) √x^4 = 9 (2) √x^2 = -x

Hi askhere,
You have to always handle roots operations carefully.
You know , In GMAT even root operation always yield +value.
For example, \(\sqrt{4}\)= +2 NOT -2 since \(\sqrt{-2}\) is imaginary and GMAT stays in real world. Only real values are accepted.
However, \(x^2=4\) has two solutions x=+2 or -2.

Now, as per above concepts, \(\sqrt{x^2}\) is always positive.
But RHS is negative of x. Negative of x will be positive when x<0 (or x is negative).--------------(1)

1) For example if x=-4, then \(LHS= \sqrt{x^2}=\sqrt{(-4)^2}=\sqrt{16}= +4\)
\(RHS=-x=-(-4)=4\)
So LHS=RHS when\(x<0\)

2) LHS will be equal to RHS when x=0

So, x can never be +ve.

From (1) and (2), \(x\leq{0}\) (More than one value of x, so insufficient)

Your method:- The moment we simplified the statement, we lost the essence of the statement ,i.e, what GMAT wanted to test us. Simplify when it is necessary or when the original statement has no worth significance. if the original statement conveys various possibilities, then we mustn't simplify.

Hope it helps.

P.S:- Never simplify root operations , first think of the logic behind the root operation, then do the needful.