Question:
Ben is climbing a tree with a lot of branches. His height off the ground at time $t$ is $2t^2-5t+29$ feet. To the nearest foot, what will his minimum height be?

Answer:
Completing the square, we get
\begin{align*}
2t^2 - 5t + 29 &= 2 \left( t^2 - \frac{5}{2} t \right) + 29 \\
&= 2 \left[ \left( t - \frac{5}{4} \right)^2 - \frac{5^2}{4^2} \right] + 29 \\
&= 2 \left( t - \frac{5}{4} \right)^2 + \frac{207}{8}.
\end{align*}Thus, the minimum height is $\frac{207}{8}.$  To the nearest integer, this is $\boxed{26}.$