What are some examples of extraneous solutions to equations?

Example 1 : Raising to an even power
Solve #x=root(4)(5x^2-4)#.
Raising both sides to the #4^(th)# gives #x^4=5x^2-4#.
This requires, #x^4-5x^2+4=0#.
Factoring gives #(x^2-1)(x^2-4)=0#.
So we need #(x+1)(x-1)(x+2)(x-2)=0#.

The solution set of the last equation is #{-1, 1, -2, 2}#. Checking these reveals that #-1# and #-2# are not solutions to the original equation. Recall that #root(4)x# means the non-negative 4th root.)

Example 2 Multiplying by zero
If you solve #(x+3)/x=5/x# by cross multiplying,
you'll get #x^2+3x=5x#
which lead to #x^2-2x=0#
.
It looks like the solution set is #{0, 2}#.
Both are solutions to the second and third equations, but #0# is not a solution to the original equation.

Example 3 : Combining sums of logarithms.
Solve: #logx+log(x+2)=log15#
Combine the logs on the left to get #log(x(x+2))=log15#
This leads to #x(x+2)=15# which has 2 solutions: #{3, -5}#. The #-5# is not a solution to the original equation because #logx# has domain #x>0# (Interval: #(0,oo)#)