Isolating a variable under square root

Question

Given this equation:

$T=\sqrt{(ugx)^2+(T_0)^2}$

You're asked to isolate $x$. My process was:

$T=ugx + T_0$ (the square root cancelled the exponents)

$T-T_0=ugx$

$x=\frac{T-T_0}{ug}$

But that was the wrong answer, and they instead followed this process:

$T^2=(\sqrt{(ugx)^2+(T_0)^2})^2$ (squaring both sides)

$T^2-(T_0)^2=(ugx)^2$

$\pm \sqrt{T^2-(T_0)^2}=\sqrt{(ugx)^2}$

$\pm \sqrt{T^2-(T_0)^2}=ugx$ (why no $\pm$ with $ugx$?)

$x=\frac{\pm \sqrt{T^2-(T_0)^2}}{ug}$

The answer further said that only positive solutions are included, so the final answer is

$x=\frac{\sqrt{T^2-(T_0)^2}}{ug}$

Doesn't that line also evaluate to

$x=\frac{T-T_0}{ug}$? I'm pretty confused.

Answer 1

Consider $$\sqrt{2^2-1^2}\ne 2-1.$$

Answer 2

Hint: Squaring we get $$T^2-T_0^2=(ugx)^2$$ so we obtain $$ugx=\pm\sqrt{T^2-T_0^2}$$ Can you finish?