Question

If, $\alpha$ and $\beta$ are the positive roots of the equation, $ax^2+bx+c=0$, find the equation whose roots are $\dfrac{1}{\sqrt\alpha}$ and $\dfrac{1}{\sqrt\beta}.$

Answer 1

Given: $\alpha,\beta$ are the positive roots of the equation $ax^2+bx+c=0$
To find: Equation with roots $\dfrac{1}{\sqrt\alpha}$ and $\dfrac{1}{\sqrt\beta}$

Let $y=\dfrac{1}{\sqrt{x}}$

$\Rightarrow x=\dfrac{1}{y^2}$

Put $x=\dfrac{1}{y^2}$ in given equation,

$a\left(\dfrac{1}{y^2}\right)^2+b\left(\dfrac{1}{y^2}\right)+c=0$

$\Rightarrow \dfrac{a}{y^4}+\dfrac{b}{y^2}+c=0$

$\Rightarrow cy^4+by^2+a=0$

Now, In terms of variables \(x,\) the transformed equation becomes:

$cx^4+bx^2+a=0$