Solve each of the following equal \[ \frac{x^{2}-5 x+6}{x^{2}-2}=0 \]

To solve the equation \[ \frac{x^{2}-5x+6}{x^{2}-2}=0, \] we need the numerator to be equal to zero since a fraction is zero when its numerator is zero, provided the denominator is not zero. Setting the numerator equal to zero: \[ x^2 - 5x + 6 = 0. \] We can factor the quadratic: \[ (x - 2)(x - 3) = 0. \] This gives us the solutions \[ x - 2 = 0 \quad \Rightarrow \quad x = 2, \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3. \] Next, we should check if the values of \(x\) we found make the denominator zero. The denominator is \[ x^2 - 2. \] Setting it to zero gives us: \[ x^2 - 2 = 0 \quad \Rightarrow \quad x^2 = 2 \quad \Rightarrow \quad x = \pm \sqrt{2}. \] The solutions \(x = 2\) and \(x = 3\) do not equal \(+\sqrt{2}\) or \(-\sqrt{2}\), so there are no restrictions from the denominator. Thus, the final solutions to the equation are \[ \boxed{2} \quad \text{and} \quad \boxed{3}. \]