## Question

###### What are some examples of extraneous solutions to equations?

## Answers

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 2Multiplying 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)# )