Quadratic equation: Which way is correct? pic1 or pic2?

[anuttarasammyak]

As the convention
$$\sqrt{}$$
denotes nonnegative square root , e.g. 2 out of 2,-2 which are square roots of 4.

That is OK but I am puzzled in using this symbol for negative or complex numbers.
For an example
$$\sqrt{-1}=i$$
Why not -i ? What is the convention ? I suppose it is "nonnegative on pure imaginary axis". Is it right?
Square roots of i are $e^{\pi/4\ i},e^{5\pi/4\ i}$. Which is $\sqrt{i}$ ?

 

[Mark44]

That is OK but I am puzzled in using this symbol for negative or complex numbers.
For an example
$$\sqrt{-1}=i$$
Why not -i ? What is the convention ? I suppose it is "nonnegative on pure imaginary axis". Is it right?
Square roots of i are $e^{\pi/4\ i},e^{5\pi/4\ i}$. Which is $\sqrt{i}$ ?

"Nonnegative on pure imaginary axis" seems to be the convention. For your second question, several web sites I looked at (search on "principal square root of a complex number") define the principal square root of a complex number as the root with a positive imaginary part. It's important to note that the imaginary part is the coefficient (a real number) of i. This would make $e^{\pi/4 i}$ the principal square root of i.

 

[anuttarasammyak]

Thank you so much @Mark44 .

So we may summarize the convention of $\sqrt{}$ that we choose from square roots
#1 nonnegative one on imaginary axis
If imaginary part are zero,
#2 nonnegative one on real axis.
Or in a word it has phase angle of $\phi[0,\pi)$.

$$\sqrt{e^{i5\pi/4}}=e^{i5\pi/8}$$
whose real part is negative and imaginary part is positive. Imaginary axis prevails.