Problem 147 - College Algebra

Question

Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.

Step-by-Step Solution

Answer

The original square's side length is 5 inches.

Step 1: Define the Problem Variables

Let \( x \) denote the length of a side of the original square in inches. Consequently, each side of the larger square is \( x+3 \) inches long.

Step 2: Establish an Equation

Given that the area of the larger square amounts to 64 square inches, and knowing that the area of a square is computed as the square of the length of one of its sides, an equation can be formulated: \( (x+3)^2 = 64 \)

Step 3: Solve the Equation

The equation \( (x+3)^2 = 64 \) simplifies to \( x^2 + 6x + 9 = 64 \). Thus, we need solve the quadratic equation \( x^2 + 6x - 55 = 0 \) by either factoring, completing the square or using the quadratic formula.

Step 4: Evaluate Solution

By factoring the quadratic equation we get \( (x - 5)(x + 11) = 0 \). Thus, the equation is satisfied for \( x = 5 \) or \( x = -11 \). However, length cannot be negative. Therefore, the side length of the original square is \( x = 5 \) inches.