Question

If $\alpha$ and $\beta$ are the roots of $x^2+px+q=0$ and ${\alpha }^4,{\beta }^4$ are the roots of $x^2-rx+5=0$, then the equation $x^2-4qx+2q^2-r=0$ has always

• A. one positive and one negative root
• B. two positive roots
• C. two negative roots
• D. cannot say anything

Answer 1

The correct option is A one positive and one negative root

Given,
$x^2+px+q=0$
$\alpha +\beta =-p$
$\alpha \beta =q$
${\alpha }^4,{\beta }^4$ are roots of $x^2-rx+5=0$
$\Rightarrow {\alpha }^4+{\beta }^4=r$
${\alpha }^4{\beta }^4=5$
$f\left(x\right)=x^2-4qx+2q^2-r=0$
$f\left(0\right)=2q^2-r$
$=2(\alpha \beta {)}^2-({\alpha }^4+{\beta }^4)$
$=-({\alpha }^2-{\beta }^2{)}^2$
$<0$
$\Rightarrow 0$ lies between the roots.
i.e., it has one positive and one negative root.
Hence, option A is correct.