Question

Solve for $x$ in the equation $2m(1+x^2)-(1+m^2)(x+m)=0$

Answer 1

$2m(1+x^2)-(1+m^2)(x+m)$

This can be written as

$\Rightarrow 2m+2m{x}^2-(1+m^2)x-m(1+m^2)=0$

$\Rightarrow 2m{x}^2-(1+m^2)x+(m-m^3)=0$

We know roots of quadratic equation $a{x}^2+bx+c$ are $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Using this result we get

$\Rightarrow x=\frac{(1+m^2)\pm \sqrt{(1+m^2{)}^2-8m(m-m^3)}}{4m}$

$\Rightarrow x=\frac{(1+m^2)\pm \sqrt{(1+m^4+2m^2)-8m^2+8m^4}}{4m}$

$\Rightarrow x=\frac{(1+m^2)\pm \sqrt{1+9m^4-6m^2}}{4m}$

$\Rightarrow x=\frac{(1+m^2)\pm \sqrt{(3m^2-1{)}^2}}{4m}$

$\Rightarrow x=\frac{(1+m^2)\pm (3m^2-1)}{4m},$

$\Rightarrow x=\frac{(1+m^2)+(3m^2-1)}{4m}$ and $x=\frac{(1+m^2)-(3m^2-1)}{4m}$

On solving we get

$\Rightarrow x=m$ and $x=\frac{1-m^2}{2m}$