If (x+\frac{2}{x}=6), what is the value of (x^2+\frac{4}{x^2})?

Correct answer: (32)

The direct answer is C: \(32\). Start with \(x+\frac{2}{x}=6\), where \(x\neq0\). Square both sides: \(\left(x+\frac{2}{x}\right)^2=6^2\). Expanding gives \(x^2+2\cdot x\cdot\frac{2}{x}+\frac{4}{x^2}=36\). The middle term simplifies to 4, so \(x^2+4+\frac{4}{x^2}=36\). Subtract 4 from both sides: \(x^2+\frac{4}{x^2}=32\). Option C is correct. Option A, 30, subtracts the wrong amount. Option B, 34, adds 2 instead of removing the middle term 4. Option D, 36, is the squared right-hand side before removing the extra middle term. The important point is that squaring creates a middle term, which must be handled. Memory cue: after squaring \(a+b\), remove \(2ab\) to obtain \(a^2+b^2\).