1. **State the problem:** Given the quadratic equation $$x^2 - 3ax + a^2 = 0$$ with roots $$\alpha$$ and $$\beta$$, and the condition $$\alpha^2 + \beta^2 = \frac{7}{9}$$, find the value of $$a$$. 2. **Recall the relationships between roots and coefficients:** For a quadratic equation $$x^2 + bx + c = 0$$ with roots $$\alpha$$ and $$\beta$$, - Sum of roots: $$\alpha + \beta = -b$$ - Product of roots: $$\alpha \beta = c$$ 3. **Apply these to our equation:** Here, the equation is $$x^2 - 3ax + a^2 = 0$$, so - $$\alpha + \beta = 3a$$ - $$\alpha \beta = a^2$$ 4. **Use the identity for sum of squares of roots:** $$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$ Substitute the known values: $$\alpha^2 + \beta^2 = (3a)^2 - 2a^2 = 9a^2 - 2a^2 = 7a^2$$ 5. **Set equal to given value and solve for $$a$$:** $$7a^2 = \frac{7}{9}$$ Divide both sides by 7: $$a^2 = \frac{1}{9}$$ Take square root: $$a = \pm \frac{1}{3}$$ 6. **Final answer:** $$a = \frac{1}{3} \text{ or } a = -\frac{1}{3}$$