1. **State the problem:** Solve the equation $x^2 - 49 = 0$ for $x$. 2. **Recall the formula:** 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 $b = 7$. 3. **Factor the equation:** $$x^2 - 49 = (x - 7)(x + 7) = 0$$ 4. **Apply the zero product property:** For the product to be zero, at least one factor must be zero: $$x - 7 = 0 \quad \text{or} \quad x + 7 = 0$$ 5. **Solve each equation:** - For $x - 7 = 0$, add 7 to both sides: $$x = 7$$ - For $x + 7 = 0$, subtract 7 from both sides: $$x = -7$$ 6. **Final answer:** The solutions to the equation are $$x = 7$$ and $$x = -7$$.