نویسنده : امیر انصاری

1. $$
   x^2+2x=k
   $$
   
2. $$
   kx^2-2x=k
   $$
   
3. $$
   x^2=kx+1
   $$
   

پاسخ

1. $$
   x^2+2x=k\\
   x^2+2x+1=k+1\\
   (x+1)^2=k+1\\
   x+1=\pm \sqrt{k+1}\\
   x=-1 \pm \sqrt{k+1}
   $$
   
2. $$
   kx^2-2x=k\\
   x^2-\frac{2}{k}x=1\\
   x^2-\frac{2}{k}x+\frac{1}{k^2}=1+\frac{1}{k^2}\\
   \bigl( x-\frac{1}{k} \bigr)^2=\frac{k^2+1}{k^2}\\
   x-\frac{1}{k} = \pm \sqrt{\frac{k^2+1}{k^2}}\\
   x=\frac{1}{k} \pm \frac{\sqrt{k^2+1}}{k}\\
   x=\frac{1 \pm \sqrt{k^2+1}}{k}
   $$
   
3. $$
   x^2=kx+1\\
   x^2-kx=1\\
   x^2-kx+\frac{k^2}{4}=1+\frac{k^2}{4}\\
   \bigl( x-\frac{k}{2} \bigr)^2 = \frac{4+k^2}{4}\\
   x-\frac{k}{2} = \pm \sqrt{\frac{4+k^2}{4}}\\
   x=\frac{k}{2} \pm \frac{\sqrt{4+k^2}}{2}\\
   x=\frac{k \pm \sqrt{4+k^2}}{2}
   $$