1. **State the problem:** Solve the quadratic equation $x^2 - 25 = 0$. 2. **Formula and rules:** This is a difference of squares equation, which can be factored using the identity $$a^2 - b^2 = (a - b)(a + b)$$ where $a = x$ and $b = 5$. 3. **Factor the equation:** $$x^2 - 25 = (x - 5)(x + 5) = 0$$ 4. **Solve for $x$ by setting each factor equal to zero:** $$x - 5 = 0 \implies x = 5$$ $$x + 5 = 0 \implies x = -5$$ 5. **Final answer:** The solutions to the equation are $$x = 5$$ and $$x = -5$$.