My answer was C, but it shows that is incorrect

is my answer wrong? if yes, then why?

or is the solution incorrect?

Hint: A is always positive. Does B have to be always positive?

Have you tested or all the cases ?

hint: test x = -6

I have a doubt, what if we simplify the quantity A as ((x-5)^{2})^{1/2} it comes out to be (x-5), so both quantities are equal.

Please correct me if I am wrong because this was my first approach to the problem.

That is true as long as x - 5 ≥ 0. Because when x < 5, quantity A reduces to (5 - x) and not (x - 5).

(x^2 -10x + 25) reduces to (x-5)^2 also (5-x)^2 -----Both are same : \

Think about it this way: Quantity A is \sqrt{x^2 - 10x + 25}. This, as you said, can be written as \sqrt{(x - 5)^2}. When x \geq 5, \sqrt{(x - 5)^2} is indeed x - 5. But when x < 5, x - 5 is negative, and hence Quantity A becomes 5 - x instead.

That is, for x < 5, the question reduces to

**Quantity A**: 5 - x

**Quantity B**: x - 5

while for x \geq 5, we get

**Quantity A**: x - 5

**Quantity B**: x - 5

You can see how **Quantity A** and **Quantity B** are not equal in the x < 5 case.

Oh, Okay got it,

Thank you for this explanation