What exactly are narrower and wider conditions? (Polya's How To Solve It) In _How To Solve It_ , on page 56, Polya states that > "If we pass from a proposed condition to a new condition equivalent to it, we have the same solutions. But if we pass from a proposed condition to a narrower one, we lose solutions, and if we pass to a wider one we admit improper, adventitious solutions which have nothing to do with the proposed problem." I know that when solving an equation, raising both sides to a power might give extraneus roots, but what other cases are there, if there are, that might give extraneus roots? And taking n-th roots might sometimes diminish the amount of solutions to the equation (the amount of solutions you get out of that particular calculation), but are there any other cases where this might happen?

If we only talk about transforming equations, then spurious solutions may appear whenever we apply a function $f$ to both sides that is not injective. In other words, we have $a=b\implies f(a)=f(b)$, but not vice versa. The case of squaring both sides is just a special case of this (with $f(x)=x^2$). Similarly, taking roots is like the attempt to go from $f(a)=f(b)$ to $a=b$; but in general that is not valid, i.e., we may lose solutions.

The effect may also occur when doing thíngs that are normally more harmless things (i.e., applying more innocent-looking $f$), such as multiplying both sides by an expression: Thus expression _may_ accidentally be zero and thereby introduce additional solutions (because multiplication by $0$ is far from injective).

Beware that even when we do not transofmr an equation as a whole, we may fall for these traps, e.g., when replacing $\sqrt{x^2}$ with $x$; this is valid only if we knwo that $x\ge0$!