Answer
$3.24\text{ hr}$
Work Step by Step
Let, $t$ represent the number of hours taken by Joel to grade the project.
The time taken by Tanner is $t+2$ hours.
In one hour, Joel does $\frac{1}{t}$ of the job and Tanner does $\frac{1}{t+2}$ of the job.
Convert to equations:
$\begin{align}
& 2\left( \frac{1}{t} \right)+2\left( \frac{1}{t+2} \right)=1 \\
& \frac{2}{t}+\frac{2}{t+2}=1
\end{align}$
Solve:
Multiply by the LCD $t\left( t+2 \right)$ on both sides of the equation,
$\begin{align}
& t\left( t+2 \right)\left( \frac{2}{t}+\frac{2}{t+2} \right)=t\left( t+2 \right)1 \\
& 4t+4=t\left( t+2 \right) \\
& 4t+4={{t}^{2}}+2t
\end{align}$
Subtract $\left( 4 \right)$ on both sides,
$\begin{align}
& 4t+4={{t}^{2}}+2t \\
& 4t+4-4={{t}^{2}}+2t-4 \\
& 4t={{t}^{2}}+2t-4
\end{align}$
Subtract $4t$ on both sides,
$\begin{align}
& 4t={{t}^{2}}+2t-4 \\
& 4t-4t={{t}^{2}}+2t-4-4t \\
& 0={{t}^{2}}+2t-4
\end{align}$
Use the quadratic formula,
$\begin{align}
& t=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-\left( -2 \right)\pm \sqrt{{{\left( -2 \right)}^{2}}-4\left( 1 \right)\left( -4 \right)}}{2\left( 1 \right)} \\
& =\frac{2\pm \sqrt{20}}{2}
\end{align}$
$\begin{align}
& \frac{2\left( 1\pm \sqrt{5} \right)}{2}=1\pm \sqrt{5} \\
& \text{ }=1+\sqrt{5}\approx 3.24 \\
& \text{ and }1-\sqrt{5}\approx -1.236 \\
\end{align}$
Therefore, the values of $t$ are $3.24,-1.236$.
Thus, it would take Joel about $3.24\text{ hr}$ to do the job.
