Answer
$11$ days
Work Step by Step
Let $t$ represent the number of days taken to plant the trees by Katherine.
Juliana takes $t+2$ hours.
In one day, Katherine does $\frac{1}{t}$ of the job and Julianna does $\frac{1}{t+2}$ of the job.
Convert to equations:
Working together, Katherine and Julianna can plant trees in $6$ days.
According to the provided statement, the equation is,
$\begin{align}
& 6\left( \frac{1}{t} \right)+6\left( \frac{1}{t+2} \right)=1 \\
& \frac{6}{t}+\frac{6}{t+2}=1
\end{align}$
Solve:
Multiply by the LCD $t\left( t+2 \right)$ on both sides of the equation,
$\begin{align}
& t\left( t+2 \right)\left( \frac{6}{t}+\frac{6}{t+2} \right)=t\left( t+2 \right)1 \\
& 6\left( t+2 \right)+6t=t\left( t+2 \right) \\
& 6t+12+6t={{t}^{2}}+2t \\
& 0={{t}^{2}}-10t-12
\end{align}$
Using the quadratic formula,
$\begin{align}
& t=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\left( 1 \right)\left( -12 \right)}}{2\left( 1 \right)} \\
& =\frac{10\pm \sqrt{100+48}}{2} \\
& =\frac{10\pm \sqrt{148}}{2}
\end{align}$
Thus, the values of t are;
$\begin{align}
& t=\frac{10+2\sqrt{37}}{2}\text{ and }t=\frac{10-2\sqrt{37}}{2} \\
& t\approx 11.082\text{ and }t\approx -1.0827 \\
\end{align}$
Thus, it would take 11 days.
