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

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Solution

let common root be $α$
so $α$ will satisfy both the equations
$α^{2} - 11 α + a = 0 - 1 α^{2} - 14 α + 2 a = 0 - 2 2 - 1 - 3 α + a = 0 α = \frac{a}{3} p u t α = \frac{a}{3} i n e q 1 \frac{a^{2}}{9} - \frac{11 a}{3} + a = 0 {3 a}^{2} - 33 a + 9 a = 0 a^{2} - 24 a = 0 a = 0 o r 24$
ubt positive value of $a$ is $24$

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