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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Sometimes, solving equations using inverse operations leads to solutions that do not correctly solve the original equation. These solutions are **extraneous**, and they occur primarily in radical equations when the radical is eliminated.
$\begin{gathered}
2=x+\sqrt{2x-1}
\end{gathered}$
This example equation has $x=1$ and $x=5$ as its solutions, but whenever an equation has multiple solutions, it's a good idea to check every one of them by substituting them back into the original equation.

$x=1$ | $x=5$ | |
---|---|---|

Substitute | $2\stackrel{?}{=}{\color{#0000FF}{1}}+\sqrt{2\cdot {\color{#0000FF}{1}}-1}$ | $2\stackrel{?}{=}{\color{#0000FF}{5}}+\sqrt{2\cdot {\color{#0000FF}{5}}-1}$ |

Simplify | $2=2$ | $2\neq 8$ |

Is valid? | Yes | No |

For the example equation, $x=1$ was a valid solution but $x=5$ was extraneous.