@Dsgrue
I think the only thing that really needs to be said is that the square root operation (like the absolute value operation) is one of those operations where two different inputs can have the same output. That's why you cannot set two values equal solely because they have the same output from these operations. Since having the same output does not imply equal input.
That is not the case with the addition, subtraction, multiplication, or division operations. Using these operations, no input will have the same output as another input. Therefore if two unknowns have the same output using these operations, then their inputs are equal. Having the same output DOES mean the inputs are the same.
The only exception is multiplying/dividing by zero, which I didn't do.
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