Quadratic Functions, Equations and Inequalities

A square has side length \(x\) cm. A rectangle has length \((x+3)\) cm and width \((x-3)\) cm, where \(x>3\).

1. Show that the square and rectangle have equal perimeters.

2. Find their areas and determine how much smaller the rectangle’s area is. Explain why equal perimeters do not guarantee equal areas.

3. Replace \(3\) by a positive number \(k\), where \(0<k<x\). Find the area difference. Hence find \(k\) if the rectangle’s area is \(25\text{ cm}^2\) less than the square’s.