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When the equations are equivalent? I know the definition: "equivalent equations - equations that have the same set of solutions.". Is that all, or do we need to add something about the domain?

For example:

1)

$(x-2)(x^2+1)=0$

$Domain:~x\in R$

$x_0=2$

2)

$\sqrt{x-2}=0$

$Domain: ~x \ge 2$

$x_0=2$

For $x = -10$, second equation doesn't exist and first equation is logical false, but both equations have the same solutions $x_0=2$. So if two equations have the same solution but have other domain, can we say that these equations are equivalent?

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    You can say they are equal for $x=2$ – 2017-02-07
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    From a "logical point of view" an equation is an open formula $\phi(x)$ that define a set: the set of "solutions" : $\{ x \mid \phi(x) \}$ (possibly : $\emptyset$). – 2017-02-07
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    Thus, we may say that two equations are "equivalent" if they define the same "solution set". – 2017-02-07

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"The same set of solutions" says it all.

$$(x-2)(x^2+1)=0\iff x=2$$

and

$$\sqrt{x-2}=0\iff x=2,$$

that's it.

The extent of the domain is implicit in the definition, as a solution is valid only if it lies in the domain.

The situation would be different if you had additional restrictions on the domain, like if you had to solve $f(x)=0$, where

$$f(x):[10,\infty)\to\mathbb R:x\to(x-2)(x^2+1)$$ (no solution).