What is x^2 -6x+9x-3+ -x if each term in this expression is correctly defined?

What is x^2 -6x+9x-3+ -x if each term in this… | GMAT Sprint

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PS · Problem Solving

705-805

Roots

What is $\frac{\sqrt{x^2 -6x+9}}{x-3}+\sqrt{-x}$ if each term in this expression is correctly defined?

Answer Choices

Correct answer marked below

\sqrt{-x}

1 + \sqrt{x}

1 + \sqrt{-x}

-1 + \sqrt{x}

-1 + \sqrt{-x}

Correct

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Explanation

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Official Explanation

Official

STEP 1: Even roots of negative numbers are not defined, thus for

\sqrt{-x}

to be defined,

-x

must be greater than or equal to 0:

-x\geq{0}

. Therefore

x\leq{0}

.

STEP 2: Recall that

\sqrt{x^2}=|x|

\sqrt{x^2 -6x+9}=\sqrt{(x-3)^2}=|x-3|

. Since from above

x\leq{0}

, then

x-3

is negative and thus

|x - 3| = -(x - 3)

.…

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What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

What is x2−6x+9x−3+−x\frac{\sqrt{x^2 -6x+9}}{x-3}+\sqrt{-x}x−3x2−6x+9​​+−x​ if each term in this expression is correctly defined?