Question

If the zero of the polynomial $x^2+4x+2a$ are $\alpha$ and $\frac{2}{\alpha }$, then find the value of $a$.

Answer 1

Comparing $x^2+4x+2a$ with $a{x}^2+bx+c$,

$a=1,b=4,c=2a$

According to the question,

$\alpha ,\frac{\alpha }{2}$ are the zeroes of the polynomial,

sum of the zeroes $\frac{b}{a}$

$\alpha +\frac{\alpha }{2}=\frac{-4}{1}$

$\alpha =\frac{-8}{3}$

${\alpha }^2=\frac{64}{9}$ [squaring both sides]

product of all zeroes,

$\alpha \times \frac{\alpha }{2}=\frac{2a}{1}{\alpha }^2=4a$

$\therefore 4a=\frac{64}{9}$ $[\because {\alpha }^2=\frac{64}{9}]$

$a=\frac{16}{9}$