Question

If $x_1,x_2$ and $x_3$ are three distinct number which satisfy the relation
$((a-1)x^2+(a^2+4a+2)x+(a^2+a-2)=0$ then

• A. $a=-1$
• B. $a=-2$
• C. $a=3$
• D. None of these

Answer 1

The correct option is D None of these

$(a-1)x^2+(a^2+4a+2)x+(a^2+a-2)=0$

$\Rightarrow a-1=0,a^2+4a+2=0,a^2+a-2=0$

$\Rightarrow a=1,a=\frac{-4\pm \sqrt{16-4\times 2\times 1}}{2},a^2+2a-a-2=0$

$\Rightarrow a=1,a=\frac{-4\pm \sqrt{8}}{2},a(a+2)-1(a+2)=0$

$\Rightarrow a=1,a=\frac{-4\pm 2\sqrt{2}}{2},(a+2)(a-1)=0$

$\Rightarrow a=1,a=-2\pm \sqrt{2},a=1,-2$

$\Rightarrow a=\left\{1\right\}\cap \{-2+\sqrt{2},-2-\sqrt{2}\}\cap \{1,-2\}$

$\therefore a\in \varphi$