Solving a usually involves three main steps.
Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.
Solve the resulting equation.
Check the results for .
Now we can analyze the given radical equation.
8 − x + 12 = 3 \begin{gathered}
8-\sqrt{x+12}=3
\end{gathered} 8 − x + 1 2 = 3
First, let's isolate the radical,
x + 12 , \sqrt{x+12}, x + 1 2 , on one side of the equation.
8 − x + 12 = 3 8-\sqrt{x+12}=3 8 − x + 1 2 = 3 - x + 12 = - 5 \text{-}\sqrt{x+12}=\text{-}5 - x + 1 2 = - 5 x + 12 = 5 \sqrt{x+12}=5 x + 1 2 = 5
We get an isolated radical with index equal to
2 . {\color{#0000FF}{2}}. 2 . Then, we will raise each side of the equation to the power of
2 . {\color{#0000FF}{2}}. 2 . x + 12 = 5 \sqrt{x+12}=5 x + 1 2 = 5 ( x + 12 ) 2 = 5 2 \left(\sqrt{x+12}\right)^2=5^2 ( x + 1 2 ) 2 = 5 2
Next, we will check for extraneous solutions. We do that by substituting
13 13 1 3 for
x x 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 − x + 12 = 3 8-\sqrt{x+12}=3 8 − x + 1 2 = 3 8 − 13 + 12 = ? 3 8-\sqrt{{\color{#0000FF}{13}}+12}\stackrel{?}{=}3 8 − 1 3 + 1 2 = ? 3 8 − 25 = ? 3 8-\sqrt{25}\stackrel{?}{=}3 8 − 2 5 = ? 3 8 − 5 = ? 3 8-5\stackrel{?}{=}3 8 − 5 = ? 3
3 = 3 ✓ 3=3\ {\color{#009600}{\Large\checkmark}} 3 = 3 ✓
Because our substitution produced a true statement we know that our answer,
x = 13 , x=13, x = 1 3 , is correct.
