The trees in a peach orchard were arranged in a square array with $N$ columns and $N$ rows. To enlarge the orchard by one row and one column to make a larger square, the owner bought and planted 29 more peach trees. How many trees are in the orchard after the enlargement?

225 trees

Explanation:

The owner previously had N x N trees, or N² trees.

Now there are N² + 29 trees, where N² + 29 is a complete square.

So we need to find out what complete square number + 29 equals another complete square number.

To have a clue what number N is, we must divide 29 by 2 = 14.5

So is 14² + 29 a complete square?

196 + 29 = 225

√225 = 15

So the new orchard has 225 trees.