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\(2x =\sqrt 5+1\)
বা, \(x=\cfrac{\sqrt5}{2}+\cfrac{1}{2}\)
\(\therefore x^2-x-1\)
\(=(\cfrac{\sqrt5}{2}+\cfrac{1}{2})^2-(\cfrac{\sqrt5}{2}+\cfrac{1}{2})-1\)
\(=\cfrac{5}{4}+\cfrac{1}{4}+\cfrac{2\times \sqrt5\times 1}{2\times 2}-\cfrac{\sqrt5}{2}-\cfrac{1}{2}-1\)
\(=\cfrac{\cancel{6}3}{\cancel{4}2}+\cfrac{\sqrt5}{2}-\cfrac{\sqrt5}{2}-\cfrac{3}{2}\)
\(=0\) (প্রমাণিত)