1. **Stating the problem:** Explain the concept of the Golden Ratio, including its mathematical definition and significance. 2. **Formula and definition:** The golden ratio $ \phi $ is defined as the positive solution to the quadratic equation $$x^2 - x - 1 = 0$$ This equation arises from the property that the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller part. 3. **Solving the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-1$. 4. **Calculate the discriminant:** $$\sqrt{(-1)^2 - 4 \times 1 \times (-1)} = \sqrt{1 + 4} = \sqrt{5}$$ 5. **Find the two roots:** $$x = \frac{1 \pm \sqrt{5}}{2}$$ Since the golden ratio is positive, we take the positive root: $$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$ 6. **Explanation:** The golden ratio $\phi$ is an irrational number with unique properties. It appears in geometry, art, architecture, and nature because it is considered aesthetically pleasing. For example, rectangles with side lengths in the ratio $\phi$ are called golden rectangles and are often used in design. 7. **Summary:** The golden ratio is the positive solution to the equation $x^2 - x - 1 = 0$, approximately equal to 1.618, and it represents a special proportional relationship found in many natural and human-made structures.