Extraneous Solution

Question

How do problems with extraneous solutions differ from problems with no real solutions or no solution at all? How are they similar? Explain your reasoning by citing examples and using mathematical vocabulary.

Answer 1

If you solve (x+3x)/x = 5/x by cross multiplying,

you'll get x²+3x=5x

factoring an x out which lead to x²−2x=0

x(x-2) = 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.

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

so -5 is an extraneous solution