Other Operations
Multiplication and division are not the only operations that can modify the solution set. For example, take the problem:
If we take the positive square root of both sides, we get:
We're not taking the square root of any negative values here, since both x2 and 4 are necessarily positive. But we've lost the solution x = −2. The reason is that x is actually not in general the positive square root of x2. If x is negative, the positive square root of x2 is -x. If the step is taken correctly, it leads instead to the equation:
This equation has the same two solutions as the original one: x = 2, and x = −2.
Read more about this topic: Extraneous And Missing Solutions
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