If the area of a certain square (expressed in square meters) is added to its perimeter (expressed in meter), the sum is 77. What is the length of a side of the square?

Answer

To solve the problem, we need to find the length of a side of a square given that the sum of its area (in square meters) and its perimeter (in meters) equals 77.

1. **Define the variable**: Let the length of a side of the square be \( x \) meters.

2. **Write the expressions for area and perimeter**:
- The area \( A \) of the square is given by:
\[ A = x^2 \quad \text{(since area of a square = side × side)} \]
- The perimeter \( P \) of the square is given by:
\[ P = 4x \quad \text{(since perimeter of a square = 4 × side)} \]

3. **Set up the equation**: According to the problem, the sum of the area and the perimeter is 77:
\[ x^2 + 4x = 77 \]

4. **Rearrange the equation**: To solve for \( x \), we rearrange the equation:
\[ x^2 + 4x - 77 = 0 \]

5. **Factor the quadratic equation**: We need to factor the quadratic equation \( x^2 + 4x - 77 \). We look for two numbers that multiply to \(-77\) and add to \(4\). The numbers \(11\) and \(-7\) satisfy this:
\[ (x + 11)(x - 7) = 0 \]

6. **Set each factor to zero**: Now we set each factor equal to zero:
\[ x + 11 = 0 \quad \Rightarrow \quad x = -11 \quad \text{(not valid since side length cannot be negative)} \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

7. **Conclusion**: The length of a side of the square is \( 7 \) meters.