Look at my comment and then at jomo's that immediately follows.
If your first step is generally correct, then you do not need to change the sign when moving an addend from one side of an
equation to the other.
Is that generally correct? Let's experiment.
[tex]3 + 7 = 10 \implies 7 = 10 + 3 \implies 7 = 13![/tex]
When you got to the step where a square
root equals a negative number, you should have known that you had made an error because square roots are non-negative by definition.
Dr. P has suggested that you made two mistakes that offset each other. I am not 100% sure I agree with that despite my great respect for him. Your negative square root was not a separate mathematical mistake, but rather a logical consequence of your first mistake. Of course it should have warned you that something was very wrong with your logic, and failure to heed a warning is a mistake. But failing to heed a warning is not a type of mistake that offsets logical mistakes.
Your question about why you nevertheless got the correct answer is an excellent one. It frequently happens that squaring an equation introduces spurious answers. You are normally told to check for such spurious answers after solving an equation by squaring. Here the squaring luckily introduced the "correct" answer. You should not count on such luck.
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