1. Stating the problem: Solve the quadratic equation $$0 = x^2 - 6x + 9$$. 2. Formula and rules: The quadratic equation is in standard form $$ax^2 + bx + c = 0$$ where $$a=1$$, $$b=-6$$, and $$c=9$$. 3. We can solve this by factoring or using the quadratic formula. Let's try factoring first. 4. Factor the quadratic: Look for two numbers that multiply to $$9$$ and add to $$-6$$. These numbers are $$-3$$ and $$-3$$. 5. So, $$x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2$$. 6. Set the factor equal to zero: $$ (x - 3)^2 = 0 $$. 7. Solve for $$x$$: $$x - 3 = 0$$ which gives $$x = 3$$. 8. This means the quadratic has one real root (a repeated root) at $$x = 3$$. Final answer: $$x = 3$$.