1. **Stating the problem:** Solve the equation $$x^2 = -9$$ for real numbers $x$. 2. **Formula and rules:** The equation is a quadratic equation of the form $$x^2 = c$$. - For real numbers, the square of any real number is always non-negative. - Therefore, if $c$ is negative, there are no real solutions. 3. **Intermediate work:** Since $$x^2 = -9$$, and $$-9 < 0$$, there is no real number $x$ such that $$x^2 = -9$$. 4. **Conclusion:** The equation has no real solutions because the square of a real number cannot be negative. **Final answer:** No real solutions.