I tried searching before posting this, but couldn't find anything. I have this parametric* equation :
$ x^2 + 54x + 5a^2 = 0 $
They are asking me to find the values of a for which the roots of the equation will be
$x_{1} = (1/5)x_{2}$
The things that I tried :
$ x_{1} + (1/5)x_{2} = (1/5)(-b/a) $
$x_{1} * (1/5)x_{2} = (1/5)(c/a)$
But I can't get to the right answer. If you don't feel like explaining just tell me what am I doing wrong.
Your strategy of using the Vieta relations works quickly.
Let the roots be $r$ and $5r$.
Then the sum of the roots is $6r$. It is also $-54$, so $r=-9$.
The product of the roots is $5(-9)^2$. It is also $5a^2$. So $a^2=81$.