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.