TITLE: How is this proof flawed?
QUESTION [6 upvotes]: $\sqrt{x}=-1$
$\sqrt{x}^2=(-1)^2$
$x=1$
Now substitute it into the original equation
$\sqrt{1}=-1$
$1=-1$

REPLY [1 votes]: The reason is that $ \sqrt x $ is defined only for positive numbers. Now, when you square it, you get $ |x| $ and not $x$, since $x$ by itself can be anything: $+ve$ or $-ve$. ( In the case of $ \sqrt x $, the very fact that the expression is valid implies $x$ is $+ve$ : you see the difference? ).
I've committed this mistake a hell lot of times, and my teacher has scolded us (I obviously wasn't the only person to repeatedly commit the same mistake) a lot of times for the same. So I know it! :D
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