guerrero wrote:Which of the following always equals √ 9+x2−6x

x−3
3+x
|3−x|
|3+x|
3−x

OA c

By definition, √(x²) = |x|.
Thus:
√(9+x²-6x) = √(3-x)² = |3-x|.

The correct answer is C.

For some test-takers, an easier approach will be to plug in.
Note that -- by definition -- √ means the POSITIVE ROOT ONLY.
Since the correct answer choice must work for ALL values of x, be sure to try EXTREME values.

If x=1, then √(9+x²-6x) = √4 = 2.
Eliminate any answer choice that does not equal 2 when x=1:

x−3 = 1-3 = -2. Eliminate A.
3+x = 3+1 = 4. Eliminate B.
|3−x| = |3-1| = 2. Hold onto C.
|3+x| = |3+1| = 4. Eliminate D.
3−x = 3-1 = 2. Hold onto E.

If x=10, then √(9+x²-6x) = √49 = 7.
Eliminate any remaining answer choice that does not equal 7 when x=10:

|3−x| = |3-10| = 7. Hold onto C.
3−x = 3-10 = -7. Eliminate E.

The correct answer is C.