1. **State the problem:** Simplify the expression $$\sqrt{6 - 2\sqrt{5}}$$ into a simpler radical form. 2. **Recall the formula:** Expressions of the form $$\sqrt{a - 2\sqrt{b}}$$ can often be rewritten as $$\sqrt{m} - \sqrt{n}$$ where $$m$$ and $$n$$ are positive numbers satisfying: $$a = m + n$$ and $$b = mn$$. 3. **Set up equations:** Let $$\sqrt{6 - 2\sqrt{5}} = \sqrt{m} - \sqrt{n}$$. Squaring both sides: $$6 - 2\sqrt{5} = (\sqrt{m} - \sqrt{n})^2 = m + n - 2\sqrt{mn}$$ Matching terms: - The rational part: $$6 = m + n$$ - The irrational part: $$2\sqrt{5} = 2\sqrt{mn} \implies \sqrt{5} = \sqrt{mn} \implies mn = 5$$ 4. **Solve the system:** From $$m + n = 6$$ and $$mn = 5$$, consider $$m$$ and $$n$$ as roots of the quadratic: $$x^2 - 6x + 5 = 0$$ Using the quadratic formula: $$x = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm \sqrt{16}}{2} = \frac{6 \pm 4}{2}$$ So, $$x_1 = \frac{6 + 4}{2} = 5$$ $$x_2 = \frac{6 - 4}{2} = 1$$ 5. **Check values:** $$m = 5, n = 1$$ 6. **Write the simplified form:** $$\sqrt{6 - 2\sqrt{5}} = \sqrt{5} - \sqrt{1} = \sqrt{5} - 1$$ **Final answer:** $$\boxed{\sqrt{5} - 1}$$