About Square Rooting

Question

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 $$ x = ± 2\quad ?$$

Answer 1

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 $x = \pm 2$.

Answer 2

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.