A gardner plants his garden with 5550 trees and arranged them so that there is one plant more per row as there are rows then numbers of trees in a row is :

Options

A. 56
B. 74
C. 76
D. 75

Answer

To solve the problem, we need to find the number of trees in each row when a gardener plants 5550 trees, arranging them such that there is one more tree per row than the number of rows.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( n \) be the number of rows. According to the problem, the number of trees in each row will be \( n + 1 \).

2. **Set Up the Equation**:
The total number of trees can be expressed as the product of the number of rows and the number of trees per row:
\[ n \times (n + 1) = 5550 \]
This simplifies to:
\[ n^2 + n - 5550 = 0 \]

3. **Solve the Quadratic Equation**:
We will use the quadratic formula to solve for \( n \):
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 1 \), and \( c = -5550 \).

4. **Calculate the Discriminant**:
First, calculate the discriminant \( b^2 - 4ac \):
\[ D = 1^2 - 4 \times 1 \times (-5550) = 1 + 22200 = 22201 \]

5. **Find the Roots**:
Now substitute back into the quadratic formula:
\[ n = \frac{-1 \pm \sqrt{22201}}{2} \]
Calculate \( \sqrt{22201} \) which is approximately \( 149.003 \):
\[ n = \frac{-1 \pm 149.003}{2} \]
This gives us two potential solutions:
\[ n = \frac{148.003}{2} \approx 74.0015 \quad \text{(taking the positive root)} \]
Since \( n \) must be a whole number, we take \( n = 74 \).

6. **Determine the Number of Trees per Row**:
Now, we can find the number of trees in each row:
\[ \text{Number of trees per row} = n + 1 = 74 + 1 = 75 \]

### Final Answer:
The number of trees in a row is **75**.

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