Explanation

(b)
Here, $\alpha + \beta = \frac{p + 1}{2}$ and $\alpha\beta = \frac{p - 1}{2}$
Now, $(\alpha - \beta)^{2} = (\alpha + \beta)^{2} - 4\alpha\beta$
$\Rightarrow (\alpha\beta)^{2} = (\alpha + \beta) - 4\alpha\beta\ \ \ \ \lbrack\because\ \alpha - \beta = \alpha\beta\ given\rbrack$
$\Rightarrow \ \left( \frac{p - 1}{2} \right)^{2} = \left( \frac{p + 1}{2} \right)^{2} - 4\left( \frac{p - 1}{2} \right)$
$\Rightarrow p^{2} + 1 - 2p = p^{2} + 1 + 2p - 8p + 8\ \ \Rightarrow \ \ p = 2$