Question

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

• A. 2 real roots
• B. 2 positive roots
• C. 2 negative roots
• D. 1 positive and 1 negative root

Answer 1

The correct option is D 1 positive and 1 negative root

Given question is flexible. So, we can assume the roots and can make an other quardatic equation.

Let $\alpha =1$ and $\beta =2$, then the expression becomes $x^2-3x+2=0$. So p = -3, q = 2, r = 17, s = 16. The new expression is $x^2-8x-9=0$. 1 root is positive and other is negative.