1. ## Curious about the plus/minus aspect of square roots

Hello,

I understand that

$\displaystyle x^2=4$

Then $\displaystyle x= (+/-)2$

But what if I have

$\displaystyle x^2 = 4y^2$

Would that be x=2(plus/minus y) or would the -2 have to be included in the plus/minus part?

2. $\displaystyle x=\pm2$ is a contraction for "x=2 or x=-2". Even though $\displaystyle \pm y$ may seem a number, like y or -y, it cannot be viewed in isolation. Usually one has some proposition (i.e., an expression that is either true or false) $\displaystyle A(\pm y)$. This means "$\displaystyle A(y)$ or $\displaystyle A(-y)$". Therefore, "$\displaystyle x=2(\pm y)$" and "$\displaystyle x=\pm 2y$" are the equivalent and mean "x=2y or x = -2y".

3. Originally Posted by Lord Darkin
Hello,

I understand that

$\displaystyle x^2=4$

Then $\displaystyle x= (+/-)2$

But what if I have

$\displaystyle x^2 = 4y^2$

Would that be x=2(plus/minus y) or would the -2 have to be included in the plus/minus part?
To be pedantic $\displaystyle x = \pm 2$ is simply wrong because a number can't be positive and negative simultaneously. Compare emakarov's reply for that (She/he didn't use "and", I know).

1. Per definition the square-root is positive.

2. $\displaystyle x^2 = 4~\implies~|x| = 2$

3. Per definition the absolute value is

$\displaystyle |x|=\left\{\begin{array}{rcl}x&if&x\geq 0\\-x&if&x<0\end{array}\right.$

4. So

$\displaystyle x^2=4y^2~\implies~|x|=\left\{\begin{array}{rcl}2y& if&2y\geq 0\\-2y&if&2y<0\end{array}\right.$

4. Originally Posted by earboth
To be pedantic $\displaystyle x = \pm 2$ is simply wrong because a number can't be positive and negative simultaneously. Compare emakarov's reply for that (She/he didn't use "and", I know).
To be even more pedantic, $\displaystyle \pm$ does not mean "plus and minus", it means "plus or minus" so that $\displaystyle x= \pm 2$, meaning "x is either 2 or -2" is perfectly correct.

1. Per definition the square-root is positive.

2. $\displaystyle x^2 = 4~\implies~|x| = 2$

3. Per definition the absolute value is

$\displaystyle |x|=\left\{\begin{array}{rcl}x&if&x\geq 0\\-x&if&x<0\end{array}\right.$

4. So

$\displaystyle x^2=4y^2~\implies~|x|=\left\{\begin{array}{rcl}2y& if&2y\geq 0\\-2y&if&2y<0\end{array}\right.$