Determine positive $a$ such that $x^{2} - 11 x + a$ and $x^{2} - 14 x + 2 a$ may have a common factor.

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Solution

Given $x^{2} - 11 x + a$ and $x^{2} - 14 x + 2 a$ have a common factor.
Let $x - α$ be the common factor.
$\Rightarrow α^{2} - 11 α + a = 0$ -----(1)
and $α^{2} - 14 α + 2 a = 0$ ------(2)
(1)-(2) gives $α = \frac{a}{3}$
substituting $α$ in (1) gives
$a^{2} - 24 a = a (a - 24) = 0$
$∴ a = 0, 24$

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