Question

For what values of $a$ does the equation $x^2+2a\sqrt{a^2-3}x+4=0$ possess equal roots?

Answer 1

The equation $x^2+2a\sqrt{a^2-3}x+4=0$ possess equal roots if discriminant is zero i.e. $D=0$

$\Rightarrow b^2-4ac=0$

$\Rightarrow 4a^2(a^2-3)-4\times 1\times 4$=0

$\Rightarrow 4a^4-12a^2-16=$

$\Rightarrow 4a^4-16a^2+4a^2-16=0$

$\Rightarrow 4a^2(a^2-4)+4(a^2-4)=0$

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

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