Answer
$t=9$
Work Step by Step
Isolating the term with a radical, the given equation, $
t+\sqrt{t}=12
,$ is equivalent to
\begin{align*}
\sqrt{t}=12-t
.\end{align*}
Squaring both sides, the equation above is equivalent to
\begin{align*}\require{cancel}
\left(\sqrt{t}\right)^2&=(12-t)^2
\\
t&=(12)^2+2(12)(-t)+(-t)^2
&(\text{use }(a+b)^2=a^2+2ab+b^2)
\\
t&=144-24t+t^2
\\
0&=t^2+(-24t-t)+144
\\
0&=t^2-25t+144
\\
t^2-25t+144&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(t-16)(t-9)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
t-16=0 & t-9=0
\\
t=16 & t=9
.\end{array}
Checking the solutions by substitution in the given equation results to
\begin{array}{l|r}
\text{If }t=16: & \text{If }t=9
\\\\
16+\sqrt{16}\overset{?}=12 &
9+\sqrt{9}\overset{?}=12
\\
16+4\overset{?}=12 &
9+3\overset{?}=12
\\
20\ne12 &
12\overset{\checkmark}=12
.\end{array}
Since $t=16$ does not satisfy the original equation, then the only solution of the equation $
t+\sqrt{t}=12
$ is $t=9$.
