Question

Which of the following constants can be added to x^2+2/3 to form a perfect square trinomial?

Answer 1

As your question is stated, there is no answer!

I'm going to assume the polynomial you meant is:

$x^2 + \frac{2}{3} x$

To create a perfect square trinomial, take half the coefficient of x and then square it.

$\left( \frac{1}{2} \times \frac{2}{3} \right) ^2 = \left(\frac{1}{3} \right) ^2 =\frac{1}{9}$

In all problems like this, you want to square half the coefficient of x -- always half.