<s>[INST] Solving a quadratic equation with parameters $$\frac { x+2 }{ a }- \frac { 2-a }{ x } =2$$
Steps I took:
$$\frac { x(x+2) }{ ax } -\frac { a(2-a) }{ ax } =\frac { 2ax }{ ax } $$
$$x^2+2x-2a+a^2=2ax$$
$$x^{ 2 }+2x-2a+a^{ 2 }-2ax=0$$
I get stuck at this point because I have no clue how to factor an equation in this form. Hints a more appreciated than the actual answer
 [/INST] Notice, we have $$\frac{x+2}{a}-\frac{2-a}{x}=2$$
$$\frac{x(x+2)-a(2-a)}{ax}=2ax$$  $$x^2+2x-a(2-a)=2ax$$
$$x^2-2(a-1)x+a(a-2)=0$$ $$x=\frac{2(a-1)\pm\sqrt{4(a-1)^2-4a(1)(a-2)}}{2(1)}$$
$$=\frac{2(a-1)\pm\sqrt{4(a-1)^2-4a(1)(a-2)}}{2(1)}=\frac{2(a-1)\pm 2}{2(1)}$$ 
$$=\frac{2(a-1)\pm 2}{2}$$ 
 $$\iff x=\frac{2(a-1)+2}{2}=a$$
$$\iff x=\frac{2(a-1)-2}{2}=a-2$$
Hence, we get
$$\bbox[5pt, border:2.5pt solid #FF0000]{\color{blue}{x=a,\ (a-2)}}$$
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