Question

The value of ${}^{\prime }{\alpha }^{\prime }$ for which the equations $x^3+\alpha x+1=0$ and $x^4+\alpha x^2+1=0$ have exactly two common roots is

• A. $-4$
• B. $-1$
• C. $\sqrt{2}$
• D. none of these

Answer 1

The correct option is C $\sqrt{2}$

Let $\beta$ be the common root of the equations
${\beta }^3+\alpha \beta +1=0.....\left(i\right)$
${\beta }^4+\alpha {\beta }^2+1=0.....\left(ii\right)$
$\left(i\right)-\left(ii\right)=0$
$\beta ({\beta }^2+\alpha )(\beta -1)=0$
$\therefore \beta =0,1,{\alpha }^{\frac{1}{2}},-\left({\alpha }^{\frac{1}{2}}\right)$
For the equations to have exactly two common roots $\alpha$ has to be non negative.