Actually it won't work theoretically!
Since you'd have to calculate brackets before anything else!
So you didn't prove it theoretically, you just ignored mathematical rules in which order you have to calculate stuff since taking the sqareroot of the square of something basically is the same as if you'd take the absolute value of the number and it doesn't matter if you have -0,5 or 0,5 both have the absolute value of 0,5. So from the step "making the root" on your calculations are wrong and not a theoretical but not practical prove.
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That is removing the root from both sides
Yes but squaring a number actually makes them lose their little sign which means minus. Therefore -0,5 and +0.5 are exactly the same after squaring them. So if you "remove" the square by taking the squareroot you'd acually have the absolute value of the number and because of that you have to be carefull, since you can't add or substract from it as you'd normaly do. You'd have to watch out because of the absolute value.
Another example
x^2=4 has the solutions 2 AND -2 but from taking the squarerroot you only get ONE answer!
So please be carefull with even squares and their roots! Don't drop an important bit of information and if you get a mathematically wrong answer like you did, you probably messed up the square and squareroot thing.