1. **State the problem:** Solve the quadratic equation $x^2 - 45 = 0$ for $x$. 2. **Formula and rules:** The equation is in the form $x^2 = c$, where $c$ is a positive number. To solve for $x$, take the square root of both sides: $$x = \pm \sqrt{c}$$ 3. **Apply the formula:** Here, $c = 45$, so $$x = \pm \sqrt{45}$$ 4. **Simplify the surd:** Factor 45 into $9 \times 5$, so $$x = \pm \sqrt{9 \times 5} = \pm \sqrt{9} \times \sqrt{5} = \pm 3\sqrt{5}$$ 5. **Approximate to 2 decimal places:** Calculate $3 \times \sqrt{5} \approx 3 \times 2.236 = 6.71$ 6. **Final answer:** $$x = \pm 3\sqrt{5} \approx \pm 6.71$$ This means the two solutions are approximately $6.71$ and $-6.71$.