Solve the quadratic equation for x (either by completing the square or using the quadratic formula):
x = (-b (+ or -) sqrt(b*b-4*a*c))/(2*a)
So,
x = 2 + sqrt(5) or x = 2-sqrt(5)
Then use algebra to find 1/x = 1/(2+sqrt(5)) which if we multiply the numerator and denominator by the conjugate (2-sqrt(5)) and simplify we get 1/x = (2-sqrt(5))/(-1). If we picked the other solution of the quadratic formula and found 1/x we would get (2+sqrt(5))/(-1).
So, either way when we add x and 1/x we get 2*sqrt(5).