Devneet
When we plug, say x=1, then sqrt(x^2-6x+9) = sqrt(4) = +2 or -2. Note that the question just says that the expression UNDER the square root should be greater than or equal to 0 which is 4 in this case and it is fine. But, the square root of 4 can be both 2 or -2 & there is no constraint in the question that says, -2 cannot be a solution of the expression. If we consider +2 as the solution, then option A holds i.e. sqrt(2-x) since, x-3=-2 & hence, +2 from the first expression & -2 from this last expression i.e. x-3 cancels out. However, if we consider -2 as the solution of the first expression, then it becomes -2 + sqrt(2-x) -2 = -4 - sqrt(2-x) which makes option B as the correct choice. Therefore, the question definitely has a flaw as per current wording as both A & B can be the possible answer choices.
This is wrong.
\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.
The graph of the function f(x) = √xNotice that it's defined for non-negative numbers and is producing non-negative results.
TO SUMMARIZE:
When the GMAT (and generally in math) provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:
\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;
Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).