Question

a square picture with a side length of 4 inches needs to be enlarged. the final area needs to be 81 square inches. which equation can be used to solve for x, the increase in side length of the square in inches? \\(\circ\\ x^{2}+4x - 81 = 0\\) \\(\circ\\ x^{2}+4x - 65 = 0\\) \\(\circ\\ x^{2}+8x - 65 = 0\\) \\(\circ\\ x^{2}+8x - 81 = 0\\)

Explanation:

Step1: Determine new side length

The original side length is 4 inches, and the increase is \( x \) inches. So the new side length is \( 4 + x \) inches.

Step2: Use area formula for square

The area of a square is \( \text{side length}^2 \). The final area is 81, so \( (4 + x)^2 = 81 \).

Step3: Expand the left - hand side

Expanding \( (4 + x)^2 \) using the formula \( (a + b)^2=a^{2}+2ab + b^{2} \), we get \( 16+8x + x^{2}=81 \).

Step4: Rearrange the equation

Subtract 81 from both sides: \( x^{2}+8x + 16-81 = 0 \), which simplifies to \( x^{2}+8x - 65 = 0 \).

Answer:

\( x^{2}+8x - 65 = 0 \) (corresponding to the option \( x^{2}+8x - 65 = 0 \))