Square Root Confusion

Question

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?

Answer 1

Note that $x = y$ if and only if $-x = -y$. Also, $x = -y$ if and only if $-x = y$. So there are really only two possibilities:

$$x = y \qquad \mbox{or} \qquad x = -y.$$

In other words, once you know that $x^2 = y^2$, then you know that $x$ and $y$ have the same magnitude (the same absolute value); you also know that either $x$ and $y$ are exactly the same number (same magnitude and same sign), or they are exactly opposite (same magnitude but opposite signs; one is positive and the other is negative).

Answer 2

If $x^2=y^2$, then $y=\pm x$, which is the same as $x=\pm y$. But $\sqrt{x^2}=|x|$.

Answer 3

$$\sqrt {x^2}= \sqrt {y^2} \implies |x|=|y| \implies \pm y=\pm x$$ https://www.desmos.com/calculator/debdx80itq