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I have read that "every positive number $a$ has two square roots, positive and negative". For that reason I have always (as far as I could remember) unconsciously done the following for such expressions

$$ x^2 = 4 \implies x= \pm 2 $$

What I wanted to know was that, in order to cancel the square, aren't we taking square root on both sides? If we are, why don't we have something like this: $$ x^2 = 4 $$ $$ \sqrt{x^2} = \sqrt4 $$ $$ \pm x = \pm 2$$

Why do we always end up with this instead $$ x = ± 2\quad ?$$

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perhaps because $\pm x = \pm 2$ means $+x=+2$ or $-x=-2$ (more restrictive!). Further $\sqrt{x^2}=|x|$. – Raymond Manzoni Jul 8 '12 at 21:43
Hmm, I think $\sqrt{2^2} = -2 \neq |2|$ is a true statement. What is true is that $|x| = (\sqrt{x})^2$. Also, judging from the OP's comments, I think in this case the OP's intention was $\pm x = \pm 2$ or $\mp x = \pm 2$. – user12014 Jul 8 '12 at 21:53
up vote 4 down vote accepted

Consider all the possibilities of $\pm x = \pm 2$:

  • $x = -2$
  • $x = 2$
  • $-x = -2$ which implies $x = 2$
  • $-x = 2$ which implies $x = -2$

so $\pm x = \pm 2$ is the same as $x = \pm 2$.

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Thanks for clearing that up!! – Rajeshwar Jul 8 '12 at 21:44

Implication $x^2=4 \implies x= \pm2$ means, that we have two solutions of quadratic equation. Sometimes it is interpreted, that we calculate square roots of left and right part of the equation. However from pedagogical point of view this interpretation is not very good and can lead to brainless manipulations with expressions.

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