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? \\(\bigcirc\\) \\(x^2 + 4x - 81 = 0\\) \\(\bigcirc\\) \\(x^2 + 4x - 65 = 0\\) \\(\bigcirc\\) \\(x^2 + 8x - 65 = 0\\) \\(\bigcirc\\) \\(x^2 + 8x - 81 = 0\\)

Explanation:

Step1: Determine the new side length

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

Step2: Use the area formula for a square

The area of a square is \( \text{side length}^2 \). The final area is 81 square inches, so we set up the equation:
\[
(4 + x)^2 = 81
\]

Step3: Expand the left - hand side

Using the formula \( (a + b)^2=a^{2}+2ab + b^{2} \), where \( a = 4 \) and \( b=x \), we have:
\[
16+8x + x^{2}=81
\]

Step4: Rearrange the equation to standard quadratic form

Subtract 81 from both sides of the equation:
\[
x^{2}+8x + 16-81 = 0
\]
Simplify the constant terms:
\[
x^{2}+8x-65 = 0
\]

Answer:

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