To solve the equation x^2 + 2/9 = 5/12 + 1/4, we can start by simplifying both sides of the equation.

On the right side, we can add 5/12 and 1/4 by finding a common denominator. The common denominator of 12 and 4 is 12, so we have:

5/12 + 1/4 = (5/12)(1) + (1/4)(3/3) = 5/12 + 3/12 = 8/12

Now, the equation becomes:

x^2 + 2/9 = 8/12

Next, we can simplify the right side further by reducing the fraction 8/12. The greatest common divisor of 8 and 12 is 4, so we can divide both the numerator and denominator by 4:

8/12 = (8/4)/(12/4) = 2/3

Now, the equation becomes:

x^2 + 2/9 = 2/3

To eliminate the fraction on the left side, we can multiply both sides of the equation by 9:

9(x^2 + 2/9) = 9(2/3)

This simplifies to:

x^2 + 2 = 6/3

x^2 + 2 = 2

Next, we can subtract 2 from both sides of the equation:

x^2 = 2 - 2

x^2 = 0

Finally, we can take the square root of both sides to solve for x:

x = ±√0

Since the square root of 0 is 0, the solution to the equation is:

x = 0