If the roots of x⁴ – 8x³ + 23x² + kx + 12 = 0 are such that the difference of the two roots is equal to the difference of other two then k =

$\mathrm{\alpha }-\mathrm{\beta }=\mathrm{\gamma }-\mathrm{\delta }\Rightarrow \mathrm{\alpha }+\mathrm{\delta }=\mathrm{\beta }+\mathrm{\gamma }\text{ proceed as in(II) }$

$$\begin{gathered} \alpha +\beta +\gamma +\delta =8\Rightarrow \alpha +\delta =\beta +\gamma =4 \\ \alpha \beta +\beta \gamma +\gamma \delta +\delta \alpha +\delta \beta +\alpha \gamma =23 \\ \alpha (\beta +\gamma )+\delta (\beta +\gamma )+\alpha \delta +\beta \gamma =23 \\ (\alpha +\delta )(\beta +\gamma )+\alpha \delta +\beta \gamma =23 \\ \alpha \delta +\beta \gamma =23-16=7 \\ \alpha \beta \gamma +\beta \gamma \delta +\gamma \delta \alpha +\delta \alpha \beta =-k \\ \Rightarrow \alpha \delta (\alpha +\delta )+\beta \gamma (\beta +\gamma )=-k \\ \Rightarrow 4(\alpha \delta +\beta \gamma )=-k \\ \Rightarrow k=-28 \end{gathered}$$