Root, Square and absolute value

Always avoid squaring an equation, because it may add extra solutions to the equation. For example, say x = 3, i.e. we definitely know that x has only one value which is equal to 3. However, if you square the equation, then you get \(x^{2} = 9\), which will have two solutions x = 3 and x = -3. An extra solution (x = -3) is getting added because of squaring.

One way to simplify this equation is:

\(\sqrt{x^{2}} = |x|\)
Also, as per the definition of absolute value, |x| = x (for \(x\geq{0}\)), and |x| = -x (for x < 0).

Hence, the \(\sqrt{x^{2}} = |x| = -x\) will hold true for negative values of x.

UPDATE: Also, please note that \(\sqrt{x^{2}} = |x|\) and \(-{\sqrt{x^{2}}} = -|x|\).

Reference: GMAT Official Guide 13th edition, page 114 (section 4.0 Math Review, sub-section 7. Powers and Roots of Numbers).