Construct the quadratic that has one root (1/3)+square root (2/3).

If one root of a quadratic is (1+sqrt2)/3, then it's
conjugate, (1-sqrt2)/3, is also a root for the
quadratic.

We'll use the formula of a
quadratic:

x^2 - Sx + P = 0

S
is the sum of the roots.

P is the product of the
roots.

Since we know both roots, we'll determine their sum
and product to create the quadratic.

x1 + x2 = S = 1/3 +
(sqrt2)/3 + 1/3 - (sqrt2)/3

We'll eliminate like
terms:

x1 + x2 = S = 2/3

The
product is:

P = x1*x2 =
(1+sqrt2)(1-sqrt2)/9

The numerator represents the
difference of 2 squares:

P =
(1-2)/9

P = -1/9

We'll create
the quadratic: x^2 - 2x/3 - 1/9 = 0

We'll multiply by 9 the
quadratic:

9x^2 - 6x - 1 =
0

The final form of the quadratic is: 9x^2 -
6x - 1 = 0.