Is it wrong to say that $100$ is solution of $\sqrt x+10=0$? Is it wrong to say that 100 is solution of $\sqrt x +10=0$?


I know that range of $\sqrt{x}$ is $[0, \infty)$ by convention. This convention is because of definition of a function. So if I consider $\sqrt x+10=0$ as a simple equation then can I say hundred is a solution to the equation?

I think I can say that because if I square both sides I get $x=100$ and also if I consider $\sqrt x+10=0$ just as a equation then there is no need of thinking about functions and the convention of omitting negative roots.
The solutions to $f(x) = k$ are a SUBSET of the solutions to $f(x)^2 = k^2$ but not all the solution to $f(x)^2 = k^2$ are solutions to $f(x) = k$.  Squaring both sides of an equation add extraneous solutions.

$\sqrt{x} + 10 = 0$

$\sqrt{x} = -10$ Doesn't just mean that $(\sqrt{x}^2 = (-10)^2$.  It ALSO means that $\sqrt{x} = -10 &lt; 0$.

When we square both sides we LOSE information.

$\sqrt{x}^2 = (-10)^2$

$x = 100$ but we have completely LOST that $\sqrt{x} &lt; 0$.

Any  $\sqrt{100} + 10 = 10 + 10 = 20 \ne 0$.  SO it simply DOESN'T work.

Consider this:

$x =2 $ has one solution. Square both sides and you get $x^2 = 4$.  Which has TWO solutions!!!  Where did that solution $x = -2$ come from?

It came because when we squared both sides we added invalid extraneous solutions.

Solutions to $x= 2$ is $\{2\}$.  Solutions to $x^2 = 4$ has solutions $\{2,-2\}$ and $\{2\} \subset \{2,-2\}$.  But it doesn't go the other way.  It only goes one way.
$√(x)+10=0$

$√(x)=-10$

$√(x)^2=(-10)^2$

$x=100$

$√(100)+10≠0$

Thus, since $√(100)+10$ does not equal zero, this equation has no solution.