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Extraneous Solution

Concept

Extraneous Solution

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. 2=x+2x1\begin{gathered} 2=x+\sqrt{2x-1} \end{gathered} This example equation has x=1x=1 and x=5x=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=1x=1 x=5x=5
Substitute 2=?1+2112\stackrel{?}{=}{\color{#0000FF}{1}}+\sqrt{2\cdot {\color{#0000FF}{1}}-1} 2=?5+2512\stackrel{?}{=}{\color{#0000FF}{5}}+\sqrt{2\cdot {\color{#0000FF}{5}}-1}
Simplify 2=22=2 282\neq 8
Is valid? Yes No

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