The area of a recangular field is `260m^(2)` . Had its length been 5 m less and the breadth 2m more, the field would have heen in the shape of a square. Find the perimeter of the field.

Answer

To solve the problem step by step, we need to find the dimensions of the rectangular field given the conditions in the question.

### Step 1: Define the variables
Let the length of the rectangular field be \( L \) meters and the breadth be \( B \) meters.

### Step 2: Set up the equations
From the problem, we know that:
1. The area of the rectangle is given by:
\[ L \times B = 260 \quad \text{(1)} \]

2. If the length is decreased by 5 meters and the breadth is increased by 2 meters, the field becomes a square. Therefore, we can write:
\[ L - 5 = B + 2 \quad \text{(2)} \]

### Step 3: Solve equation (2) for one variable
From equation (2), we can express \( L \) in terms of \( B \):
\[ L = B + 7 \quad \text{(3)} \]

### Step 4: Substitute equation (3) into equation (1)
Now, substitute \( L \) from equation (3) into equation (1):
\[ (B + 7) \times B = 260 \]
Expanding this, we get:
\[ B^2 + 7B = 260 \]
Rearranging gives us a standard quadratic equation:
\[ B^2 + 7B - 260 = 0 \quad \text{(4)} \]

### Step 5: Solve the quadratic equation (4)
To solve the quadratic equation \( B^2 + 7B - 260 = 0 \), we can use the quadratic formula:
\[ B = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 7 \), and \( c = -260 \).

Calculating the discriminant:
\[ D = b^2 - 4ac = 7^2 - 4 \times 1 \times (-260) = 49 + 1040 = 1089 \]
Now, substituting back into the quadratic formula:
\[ B = \frac{-7 \pm \sqrt{1089}}{2 \times 1} \]
Calculating \( \sqrt{1089} = 33 \):
\[ B = \frac{-7 \pm 33}{2} \]
This gives us two possible values for \( B \):
1. \( B = \frac{26}{2} = 13 \)
2. \( B = \frac{-40}{2} = -20 \) (not valid since breadth cannot be negative)

Thus, \( B = 13 \) meters.

### Step 6: Find the length using equation (3)
Now, substitute \( B = 13 \) back into equation (3):
\[ L = 13 + 7 = 20 \text{ meters} \]

### Step 7: Calculate the perimeter
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + B) = 2(20 + 13) = 2 \times 33 = 66 \text{ meters} \]

### Final Answer
The perimeter of the field is \( 66 \) meters.

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