# How do extraneous solutions arise?

##### 1 Answer
Mar 8, 2015

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: $\left\{7\right\}$.

Square both sides of $x = 7$ to get the new equation: ${x}^{2} = 49$. The solution set of this new equation is; $\left\{- 7 , 7\right\}$. The $- 7$ is an extraneous solution introduced by squaring the two expressions

Square both sides of $x + 2 = 9$ to get the new equation: ${x}^{2} + 4 x + 4 = 81$. Solve the new equation:
${x}^{2} + 4 x - 77 = 0$ so $\left(x - 7\right) \left(x + 11\right) = 0$ whose solution set is $\left\{7 , - 11\right\}$. The $- 11$ does not solve the original equation.