Question

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

Answer 1

let common root be $\alpha$

so $\alpha$ will satisfy both the equations

${\alpha }^2-11\alpha +a=0-1{\alpha }^2-14\alpha +2a=0-22-1-3\alpha +a=0\alpha =\frac{a}{3}put\alpha =\frac{a}{3}ineq1\frac{a^2}{9}-\frac{11a}{3}+a=0{3a}^2-33a+9a=0a^2-24a=0a=0or24$

ubt positive value of $a$ is $24$