Question

Suppose $p,q,r,s\in R$ and $\alpha ,\beta$ be the roots of $x^2+px+q=0$ and ${\alpha }^4,{\beta }^4$ be the roots of $x^2-rx+s=0$. If $|\alpha |\ne |\beta |$ then the equation $x^2-4qx+2q^2-r=0$ has always

• A. two imaginary roots.
• B. two positive roots.
• C. two negative roots.
• D. one positive and one negative root.

Answer 1

The correct option is D one positive and one negative root.

$\alpha ,\beta$ are the roots of $x^2+px+q=0$
$\Rightarrow \alpha +\beta =-p,\alpha \cdot \beta =q$

${\alpha }^4,{\beta }^4$ are the roots of $x^2-rx+s=0$
$\Rightarrow {\alpha }^4+{\beta }^4=r,{\alpha }^4\cdot {\beta }^4=s$

Now for the equation
$x^2-4qx+2q^2-r=0$
$\Delta =16q^2-4(2q^2-r)$
$=4(2q^2+r)$
$=4[2{\alpha }^2{\beta }^2+{\alpha }^4+{\beta }^4]$
$\Delta =4({\alpha }^2+{\beta }^2{)}^2$
$\text{Roots}=\frac{4q\pm \sqrt{4({\alpha }^2+\beta {)}^2}}{2}$
$=2\alpha \beta \pm ({\alpha }^2+{\beta }^2)$
$=(\alpha +\beta {)}^2,-(\alpha -\beta {)}^2$
Hence, one positive and one negative root.