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?

Answer 1

Let's denote the number of columns and rows in the original peach orchard as \( N \). Therefore, the total number of trees in the original orchard can be expressed as:

\[ N^2 \]

When the owner enlarges the orchard by adding one additional row and one additional column, the dimensions become \( N+1 \) for both the width and the height. Consequently, the new total number of trees becomes:

\[ (N+1)^2 \]

The problem states that 29 additional trees were bought and planted during this enlargement. This gives us the relationship:

\[ (N+1)^2 = N^2 + 29 \]

We can expand the left-hand side of the equation:

\[ N^2 + 2N + 1 = N^2 + 29 \]

Now, we can simplify this equation by subtracting \( N^2 \) from both sides:

\[ 2N + 1 = 29 \]

Next, we isolate \( N \) by subtracting 1 from both sides:

\[ 2N = 28 \]

Dividing both sides by 2, we find:

\[ N = 14 \]

Now that we know \( N \), we can determine the original number of trees in the orchard:

\[ N^2 = 14^2 = 196 \]

After the enlargement, the new total number of trees is calculated as follows:

\[ (N+1)^2 = 15^2 = 225 \]

Thus, the total number of trees in the orchard after the enlargement is:

\[ \boxed{225} \]