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## Solving Radical Equations 1.18 - Solution

Solving a radical equation usually involves three main steps.

1. Isolate the radical on one side of the equation.
2. Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.
3. Solve the resulting equation.
4. Check the results for extraneous solutions.
Now we can analyze the given radical equation. $\begin{gathered} 8-\sqrt{x+12}=3 \end{gathered}$ First, let's isolate the radical, $\sqrt{x+12},$ on one side of the equation.
$8-\sqrt{x+12}=3$
$\text{-}\sqrt{x+12}=\text{-}5$
$\sqrt{x+12}=5$
We get an isolated radical with index equal to ${\color{#0000FF}{2}}.$ Then, we will raise each side of the equation to the power of ${\color{#0000FF}{2}}.$
$\sqrt{x+12}=5$
$\left(\sqrt{x+12}\right)^2=5^2$
Solve for $x$
$x+12=5^2$
$x+12=25$
$x=13$
Next, we will check for extraneous solutions. We do that by substituting $13$ for $x$ into the original equation. If the substitution produces a true statement, we know that our answer is correct. If it does not, then it is an extraneous solution.
$8-\sqrt{x+12}=3$
$8-\sqrt{{\color{#0000FF}{13}}+12}\stackrel{?}{=}3$
Simplify
$8-\sqrt{25}\stackrel{?}{=}3$
$8-5\stackrel{?}{=}3$
$3=3\ {\color{#009600}{\Large\checkmark}}$
Because our substitution produced a true statement we know that our answer, $x=13,$ is correct.