|x|

We say $\sqrt{4}$ = $\pm2$ but it is also true that $\sqrt{x^2}$ =|x|. How is it possible that both of the above statements are true simultaneously?

Please help me.

Easy Math Editor

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Comments

One must take note that yes, for every positive real number, there are two square roots with the same magnitude but different polarity (same numerical value but one is positive and one is negative).

The square root sign (radical sign) refers to the positive square root only. So actually \sqrt{4} = 2, and in fact, ±\sqrt{4} = ±2.

This also applies to \sqrt{[x]^[2]} . Since \sqrt{[x]^[2]} cannot be negative, it has to be |x|. Even if the value of x is negative, \sqrt{[x]^[2]} = |x|.

This is where people have misconceptions. Since x ≠ |x| for negative x, \sqrt{[x]^[2]} ≠ x.

Quince Pan - 6 years ago

Oops, I'm not used to LaTeX in Brilliant. Help pl0x...

Quince Pan - 6 years ago

Thanks for the solution.......You may check the formatting guide provided by Brilliant or may read about LaTeX on Wikipedia

Samarth Agarwal - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$