The number of integral solutions of the equation \(7\left(y+\frac{1}{y}\right)-2\left(y^2+\frac{1}{y^2}\right)=9 \text { is }\)

Let t = y+1/y; then y^2+1/y^2 = t^2-2, so 7t - 2t^2 + 4 = 9, i.e. 2t^2 - 7t + 5 = 0, giving t = 1 or 5/2. t = 1 has no real y; t = 5/2 gives y = 2 or 1/2. Only y = 2 is an integer: 1 solution.