If [a + (1/a)] 2– 2[a – (1/a)] = 12, then which of the following is a value of 'a'?
None of these
Substitute \(y=a-\tfrac{1}{a}\), so \(\left(a+\tfrac{1}{a}\right)^2 = y^2+4\). The equation becomes \(y^2+4-2y=12\), i.e. \(y^2-2y-8=0\), giving \(y=4\) or \(y=-2\).
Solve \(a-\tfrac{1}{a}=4 \Rightarrow a^2-4a-1=0 \Rightarrow a=2\pm\sqrt{5}\); and \(a-\tfrac{1}{a}=-2 \Rightarrow a=-1\pm\sqrt{2}\).
None of \(-8\pm\sqrt{3}\) or \(-8+\sqrt{5}\) match. Hence None of these.
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