If $p + \dfrac{1}{p} = \dfrac{5}{2} \text{ then }p^2 + \dfrac{1}{p^2}$ =

1. $\dfrac{25}{4}$

2. $\dfrac{19}{4}$

3. $\dfrac{9}{4}$

4. $\dfrac{17}{4}$

Given,

$p + \dfrac{1}{p} = \dfrac{5}{2}$

Using identity,

$\Rightarrow \Big(p + \dfrac{1}{p}\Big)^2 = p^2 + \dfrac{1}{p^2} + 2 \\[1em] \Rightarrow \Big(\dfrac{5}{2}\Big)^2 = \Big(p^2 + \dfrac{1}{p^2}\Big) + 2 \\[1em] \Rightarrow \dfrac{25}{4} = \Big(p^2 + \dfrac{1}{p^2}\Big) + 2 \\[1em] \Rightarrow \dfrac{25}{4} - 2 = \Big(p^2 + \dfrac{1}{p^2}\Big) \\[1em] \Rightarrow \Big(\dfrac{25 - 8}{4}\Big) = \Big(p^2 + \dfrac{1}{p^2}\Big) \\[1em] \Rightarrow \Big(p^2 + \dfrac{1}{p^2}\Big) = \dfrac{17}{4} \\[1em]$

Hence, Option 4 is the correct option.