Question

What are some examples of extraneous solutions to equations?

Answer 1

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)`)