Let's denote the side length of the original square as \( s \). The area of the original square is given by:
\[ \text{Area of square} = s^2 \]
After altering the dimensions, the new dimensions of the resulting rectangle are:
- One side is increased by 5 meters: \( s + 5 \)
- The other side is increased by 3 meters: \( s + 3 \)
The area of the rectangle is given to be 99 square meters, so we have the equation:
\[ (s + 5)(s + 3) = 99 \]
Expanding the left side:
\[ s^2 + 3s + 5s + 15 = 99 \] \[ s^2 + 8s + 15 = 99 \]
Now, subtract 99 from both sides to set the equation to 0:
\[ s^2 + 8s + 15 - 99 = 0 \] \[ s^2 + 8s - 84 = 0 \]
Next, we can factor the quadratic equation. We need two numbers that multiply to \(-84\) and add to \(8\). These numbers are \(14\) and \(-6\):
\[ (s + 14)(s - 6) = 0 \]
Setting each factor to zero gives us:
\[ s + 14 = 0 \quad \Rightarrow \quad s = -14 \quad \text{(not valid, as side length cannot be negative)} \] \[ s - 6 = 0 \quad \Rightarrow \quad s = 6 \]
Thus, the side length of the original square is \( s = 6 \) meters. The area of the original square is:
\[ s^2 = 6^2 = 36 \text{ square meters} \]
Therefore, the area of the original square is:
\[ \boxed{36} \text{ square meters} \]
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