1. **State the problem:** Given $a=5+2\sqrt{5}$ and $b=\frac{1}{a}$, find $a^2+b^2$. 2. **Recall the formula:** We want to find $a^2 + b^2$. Since $b=\frac{1}{a}$, this becomes $a^2 + \left(\frac{1}{a}\right)^2 = a^2 + \frac{1}{a^2}$. 3. **Use the identity:** Note that $a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2$. 4. **Calculate $a + \frac{1}{a}$:** First, find $\frac{1}{a}$: $$\frac{1}{a} = \frac{1}{5+2\sqrt{5}}$$ Rationalize the denominator: $$\frac{1}{5+2\sqrt{5}} \times \frac{5-2\sqrt{5}}{5-2\sqrt{5}} = \frac{5-2\sqrt{5}}{(5)^2 - (2\sqrt{5})^2} = \frac{5-2\sqrt{5}}{25 - 4 \times 5} = \frac{5-2\sqrt{5}}{25 - 20} = \frac{5-2\sqrt{5}}{5}$$ Simplify: $$\frac{1}{a} = 1 - \frac{2}{5}\sqrt{5}$$ 5. **Sum $a + \frac{1}{a}$:** $$a + \frac{1}{a} = (5 + 2\sqrt{5}) + \left(1 - \frac{2}{5}\sqrt{5}\right) = 6 + \left(2 - \frac{2}{5}\right)\sqrt{5} = 6 + \frac{10}{5} - \frac{2}{5} \sqrt{5} = 6 + \frac{8}{5}\sqrt{5}$$ Correcting the calculation of the coefficient: $$2 - \frac{2}{5} = \frac{10}{5} - \frac{2}{5} = \frac{8}{5}$$ So: $$a + \frac{1}{a} = 6 + \frac{8}{5}\sqrt{5}$$ 6. **Calculate $\left(a + \frac{1}{a}\right)^2$:** $$\left(6 + \frac{8}{5}\sqrt{5}\right)^2 = 6^2 + 2 \times 6 \times \frac{8}{5}\sqrt{5} + \left(\frac{8}{5}\sqrt{5}\right)^2 = 36 + \frac{96}{5}\sqrt{5} + \frac{64}{25} \times 5$$ Simplify the last term: $$\frac{64}{25} \times 5 = \frac{64}{5}$$ So: $$\left(a + \frac{1}{a}\right)^2 = 36 + \frac{96}{5}\sqrt{5} + \frac{64}{5} = 36 + \frac{64}{5} + \frac{96}{5}\sqrt{5}$$ Convert 36 to fifths: $$36 = \frac{180}{5}$$ Sum the constants: $$\frac{180}{5} + \frac{64}{5} = \frac{244}{5}$$ So: $$\left(a + \frac{1}{a}\right)^2 = \frac{244}{5} + \frac{96}{5}\sqrt{5}$$ 7. **Calculate $a^2 + b^2$:** $$a^2 + b^2 = \left(a + \frac{1}{a}\right)^2 - 2 = \left(\frac{244}{5} + \frac{96}{5}\sqrt{5}\right) - 2 = \frac{244}{5} - 2 + \frac{96}{5}\sqrt{5}$$ Convert 2 to fifths: $$2 = \frac{10}{5}$$ Subtract: $$\frac{244}{5} - \frac{10}{5} = \frac{234}{5}$$ Final answer: $$a^2 + b^2 = \frac{234}{5} + \frac{96}{5}\sqrt{5}$$ **Answer:** $a^2 + b^2 = \frac{234}{5} + \frac{96}{5}\sqrt{5}$