Question

If the equation $x^2+2 \alpha x+\alpha^2-1=0$ and $x^2+2 \beta x+\beta^2-1=0$ have a common root $(\alpha \neq \beta)$ then the value of the expression $2 \alpha^2-4 \alpha \beta-|\alpha-\beta|+|\alpha-\beta| \beta^2$, is

Answer 1

Subtracting the two equation we get the common root as $x=-\frac{1}{2}(\alpha+\beta)$. Substituting this in any equation we get
$ \frac{1}{4}(\alpha+\beta)^2-\frac{2 \alpha(\alpha+\beta)}{2}+\alpha^2-1=0$
$\Rightarrow \frac{1}{4}(\alpha+\beta)^2-\alpha \beta-1=0$
$\Rightarrow \frac{1}{4}\left[(\alpha+\beta)^2-4 \alpha \beta\right]-1=0$
$\Rightarrow \frac{1}{4}\left[(\alpha-\beta)^2\right]=1 \Rightarrow |\alpha-\beta|=2$
$\text { now } 2 \alpha^2-4 \alpha \beta-|\alpha-\beta|+|\alpha-\beta| \beta^2$
$ =2 \alpha^2-4 \alpha \beta+2 \beta^2-2$
$=2(\alpha-\beta)^2-2=2 \cdot 4-2=8-2=6$