Answer
$p=16$
Work Step by Step
Isolating the term with a radical, the given equation, $
p-2\sqrt{p}=8
,$ is equivalent to
\begin{align*}
p-8=2\sqrt{p}
.\end{align*}
Squaring both sides, the equation above is equivalent to
\begin{align*}\require{cancel}
(p-8)^2&=\left(2\sqrt{p}\right)^2
\\
(p)^2+2(p)(-8)+(-8)^2&=4(p)
\\
p^2-16p+64&=4p
\\
p^2+(-16p-4p)+64&=0
\\
p^2-20p+64&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(p-16)(p-4)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
p-16=0 & p-4=0
\\
p=16 & p=4
.\end{array}
Checking the solutions by substitution in the given equation results to
\begin{array}{l|r}
\text{If }p=16: & \text{If }p=4
\\\\
16-2\sqrt{16}\overset{?}=8 &
4-2\sqrt{4}\overset{?}=8
\\
16-2(4)\overset{?}=8 &
4-2(2)\overset{?}=8
\\
16-8\overset{?}=8 &
4-4\overset{?}=8
\\
8\overset{\checkmark}=8 &
0\ne8
.\end{array}
Since $p=4$ does not satisfy the original equation, then the only solution of the equation $
p-8=2\sqrt{p}
$ is $p=16$.
