Question

If one zero of the polynomial $p\left(x\right)=(a^2+9)x^2+45x+6a$ is reciprocal of the other, find the value of a.

Answer 1

Given equation is $(a^2+9)x^2+45x+6a$

Lets zeroes be $k$ and $\frac{1}{K}$

Then sum of zeroes $k+\frac{1}{K}=\frac{-b}{a}=\frac{-45}{(a^2+9)}$

And product of zeroes $k\times \frac{1}{k}=\frac{c}{a}=\frac{6a}{(a^2+9)}$

$\Rightarrow \frac{6a}{(a^2+9)}=1$

$\Rightarrow a^2+9=6a$

$\Rightarrow a^2-6a+9=0$

$\Rightarrow (a{)}^2-2(a\left)\right(3)+(3{)}^2=0$

$\Rightarrow (a-3{)}^2=0$

$\Rightarrow a-3=0$

$\Rightarrow a=3$