Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha ,\beta ,\gamma ,\delta$ be four roots

$\alpha \beta =-32$

$\alpha \beta \gamma \delta =-1984\Rightarrow \gamma \delta =62$ ${\mathrm{x}}^4-18{\mathrm{x}}^3+{\mathrm{kx}}^2+200\mathrm{x}-1984=(\mathrm{x}-\alpha )(\mathrm{x}-\beta )(\mathrm{x}-\gamma )(\mathrm{x}-\delta )$ $\equiv \left({\mathrm{x}}^2-(\alpha +\beta )\mathrm{x}-32\right)\left({\mathrm{x}}^2-(\gamma +\delta )\mathrm{x}+62\right)$ $\alpha +\beta =p\equiv \left(x^2-px-32\right)\left(x^2-qx+62\right)$ $\gamma +\delta =\mathrm{q}$

equaling co-eff. of $x^3,x^2,x$

$$p+q=18.\dots \dots \left(1\right)$$

$$-62p+32q=200.\dots \dots \left(2\right)$$

$$\begin{array}{rlrl}& & & k=62+pq-32\end{array}.\dots \dots \dots \left(3\right)$$

from (i) and (ii) $p=4,q=14$

from (iii) $\mathrm{k}=86$.