If $x^{2} + p x + q = 0$ and $x^{2} + q x + p = 0$ have a common root, then

p = q

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q + p + 1 = 0

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p + q = 0

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p - q = 0

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Solution

The correct options are
A $q + p + 1 = 0$
B $p = q$
C $p - q = 0$
Let $α$ be the common root
$∴ α^{2} + p α + q = 0$ ...........(1)
and $α^{2} + q α + p = 0$ ........ (2)
Solving (1) & (2), we get,
$\frac{α^{2}}{p^{2} - q^{2}} = \frac{α}{q - p} = \frac{1}{q - p}$
$∴ α = \frac{p^{2} - q^{2}}{q - p}$ and $α = 1$
$\Rightarrow \frac{p^{2} - q^{2}}{q - p} = 1$
$\Rightarrow p^{2} - q^{2} = q - p$
(or) $(p^{2} - q^{2}) + (p - q) = 0$
$\Rightarrow (p - q) [p + q + 1] = 0$
$\Rightarrow p - q = 0$ or $p + q + 1 = 0$

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