1. **State the problem:** Simplify and solve the equation $$\frac{2p}{5} + \frac{5}{p} = 3$$. 2. **Identify the formula and rules:** To solve this, we need to combine the fractions on the left side by finding a common denominator, then solve the resulting equation. 3. **Find a common denominator:** The denominators are 5 and $p$, so the common denominator is $5p$. 4. **Rewrite each term with the common denominator:** $$\frac{2p}{5} = \frac{2p \times p}{5 \times p} = \frac{2p^2}{5p}$$ $$\frac{5}{p} = \frac{5 \times 5}{p \times 5} = \frac{25}{5p}$$ 5. **Combine the fractions:** $$\frac{2p^2}{5p} + \frac{25}{5p} = \frac{2p^2 + 25}{5p}$$ 6. **Set the equation:** $$\frac{2p^2 + 25}{5p} = 3$$ 7. **Multiply both sides by $5p$ to clear the denominator:** $$\cancel{5p} \times \frac{2p^2 + 25}{\cancel{5p}} = 3 \times 5p$$ $$2p^2 + 25 = 15p$$ 8. **Rearrange to standard quadratic form:** $$2p^2 - 15p + 25 = 0$$ 9. **Use the quadratic formula:** $$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-15$, $c=25$. 10. **Calculate the discriminant:** $$b^2 - 4ac = (-15)^2 - 4 \times 2 \times 25 = 225 - 200 = 25$$ 11. **Calculate the roots:** $$p = \frac{15 \pm \sqrt{25}}{4} = \frac{15 \pm 5}{4}$$ 12. **Find the two solutions:** $$p_1 = \frac{15 + 5}{4} = \frac{20}{4} = 5$$ $$p_2 = \frac{15 - 5}{4} = \frac{10}{4} = \frac{5}{2}$$ **Final answer:** $$p = 5 \text{ or } p = \frac{5}{2}$$