Question

The sum of the area and perimeter of a square gives a value of 15.find the perimeter of the square

Answer 1

Let the side length of the square be \( s \).

The area \( A \) of the square is given by:
\[
A = s^2
\]

The perimeter \( P \) of the square is given by:
\[
P = 4s
\]

According to the problem, the sum of the area and the perimeter is equal to 15:
\[
s^2 + 4s = 15
\]

Rearranging this equation gives us a standard quadratic equation:
\[
s^2 + 4s - 15 = 0
\]

To solve this quadratic equation, we can factor it. We look for two numbers that multiply to \(-15\) and add to \(4\). These numbers are \(5\) and \(-3\). Thus, we can factor the equation as:
\[
(s + 5)(s - 3) = 0
\]

Setting each factor to zero gives us the solutions:
\[
s + 5 = 0 \quad \Rightarrow \quad s = -5 \quad (\text{not valid as side length cannot be negative})
\]
\[
s - 3 = 0 \quad \Rightarrow \quad s = 3 \quad (\text{valid solution})
\]

Now, we can find the perimeter:
\[
P = 4s = 4 \times 3 = 12
\]

So, the perimeter of the square is \(\boxed{12}\).