Problem 148 - Algebra and Trigonometry

Question

Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.

Step-by-Step Solution

Answer

The side length of the original square is 4 inches.

Step 1: Express the new side length in terms of the original

Let's name the side length of the original square as \( x \) (in inches). According to the problem, each side of the square is lengthened by 2 inches. This means the length of the side of the new square is \( x + 2 \) inches.

Step 2: Express the area of the new square in terms of \( x \)

The area of a square is given by the square of its side length. Therefore, the area \( A \) of the new square is \((x + 2)^2\) square inches.

Step 3: Set up the equation and solve for x

We know from the problem statement that \( A = 36 \) square inches. Therefore, we can set up the equation \((x + 2)^2 = 36\). Expanding the left side of the equation we get \(x^2 + 4x + 4 = 36\). After rearranging this equation, we get \(x^2 + 4x - 32 = 0\). Solving this quadratic equation for \( x \), we get \( x = 4 \) and \( x = -8 \). Since the length cannot be a negative value, we discard \( x = -8 \) and accept \( x = 4 \) inches as the side length of the original square.