To find \(x^2 + \frac{1}{x^2}\), we square the given equation: \(\left(x + \frac{1}{x}\right)^2 = 15^2\). Expanding: \(x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 225\). This simplifies to \(x^2 + 2 + \frac{1}{x^2} = 225\). Therefore, \(x^2 + \frac{1}{x^2} = 225 - 2 = 223\).