Question

How do extraneous solutions arise?

Answer 1

In general, extraneous solutions arise when we perform non-invertible operations on both sides of an equation. (That is, they sometimes arise, but not always.)

Non-invertible operations include: raising to an even power (odd powers are invertible), multiplying by zero, and combining sums and differences of logarithms.

Example :
The equations: `x+2=9` and `x=7`, have exactly the same set of solutions. Namely: `{7}`.

Square both sides of `x=7` to get the new equation: `x^2=49`. The solution set of this new equation is; `{-7, 7}`. The `-7` is an extraneous solution introduced by squaring the two expressions

Square both sides of `x+2=9` to get the new equation: `x^2+4x+4=81`. Solve the new equation:
`x^2+4x-77=0` so `(x-7)(x+11)=0` whose solution set is `{7,- 11}`. The `-11` does not solve the original equation.