The roots of a quadratic equation, if irrational, are found as pairs. Here we have one roots as 1/3 + sqrt 2/3, the other root will be 1/3 - sqrt 2/3

We can determine the quadratic equation by multiplying the factors,

(x - 1/3 - sqrt 2/3)(x - 1/3 + sqrt 2/3)

=> (x - 1/3)^3 - (sqrt 2/3)^2

=> x^2 + 1/9 - 2x/3 - 2/3

=> x^2 - 2x/3 - 5/9

multiply all terms by 9

=> 9x^2 - 6x - 5 = 0

**The required quadratic equation is 9x^2 - 6x - 5 = 0**