If the equations $x^{2} + p x + q = 0$ and $x^{2} + p_{1} x + q_{1} = 0$ have a common roots, show that it must be either $\frac{p q_{1} - p_{1} q}{q - q_{1}}$ or $\frac{q - q_{1}}{p_{1} - p}$

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Solution

$\begin{matrix} x^{2} + p x + q = 0........... (1), x^{2} + p^{'} x + q^{'} = 0................ (2) e q u a t i o n (1) - (2) (p - p^{'}) x + (q - q^{'}) = 0 x = (\frac{q - q^{'}}{p - p^{'}}) (q - q^{'}) x^{2} = (p q^{'} - p^{'} q) x x = (\frac{p q^{'} - p^{'} q}{q - q^{'}}) p o v e d . \end{matrix}$

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