Why is the answer to a square root problem always positive and negative?

SAMPLE:

sqrt{16} = -4 and +4

Why two answers?

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- Jul 18th 2008, 05:45 AMmagentaritaSquare Root
Why is the answer to a square root problem always positive and negative?

SAMPLE:

sqrt{16} = -4 and +4

Why two answers? - Jul 18th 2008, 05:58 AMgalactus
Because $\displaystyle (-4)^{2}=16$ and $\displaystyle 4^{2}=16$

- Jul 18th 2008, 05:59 AMSimplicity
They both lead to the same answer when squared...

$\displaystyle (-4)^2 = 16$

$\displaystyle 4^2 = 16$

$\displaystyle \implies \sqrt{16} = \pm 4$

Visually it may appear clearer, consider the quadratic graph of $\displaystyle y = x^2$ (Shown Below), for every $\displaystyle \pm x$ value, it is mapped onto the same $\displaystyle y$ value. - Jul 18th 2008, 06:59 AMmagentaritaVery good reply...
I thank both replies. I want to thank

**Air**for the picture, which makes the answer clearer.

By the way, does the same applies to variables?

For exmaple:

sqrt{x^2} = -x and +x??? True or false? - Jul 18th 2008, 09:37 AMSimplicity
- Jul 18th 2008, 09:37 AMPlato
I agree that there are

**two square roots**of any__positive__real number.

But, I disagree completely that $\displaystyle \sqrt {16} = \pm 4$. That is simply an abuse of notation!

This is a standard discussion in any elementary mathematics course.

The two square roots of 16 are: $\displaystyle \sqrt {16} = 4\,\& \, - \sqrt {16} = - 4$.

Therefore, $\displaystyle \sqrt {x^2 } = \left| x \right|$. - Jul 18th 2008, 10:31 AMMoo
Completely agree with that !

If you see the graph of the function y=sqrt(x), you will see that y can't be negative.

Actually, working with the graph y=x² is a mistake because sqrt(x) is not the inverse function of x² !!!!

If one has x²=16, then for sure x=+ or - sqrt(16) because x²=16 --> x²-16=0 --> x²-(sqrt(16))²=0 --> (x-sqrt(16))(x+sqrt(16))=0 and the rest follows.

Actually, I'd say that these two messages are contradictory. You state clearly that sqrt(x²)=|x|, which is true.

So since 16=(-4)²=4², sqrt(16)=|4|=|-4|=4 ! ;) - Jul 27th 2008, 11:13 AMmagentaritaFabulous work!
I thank all those who took time to help me understand this concept more and more.