Discuss the nature of the roots of $$ -x^{2}+2 x=-2 $$

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1. Rewrite the equation in standard form:
   $$
   -x^{2}+2x=-2 \quad \Longrightarrow \quad -x^{2}+2x+2=0.
   $$

2. Multiply both sides by $$-1$$ to simplify:
   $$
   x^{2}-2x-2=0.
   $$

3. Identify the coefficients from $$ ax^2+bx+c=0 $$:
   $$
   a=1,\quad b=-2,\quad c=-2.
   $$

4. Compute the discriminant:
   $$
   \Delta = b^{2}-4ac=(-2)^{2}-4(1)(-2)=4+8=12.
   $$

5. Since $$\Delta>0$$, there are two distinct real roots.

6. Find the roots using the quadratic formula:
   $$
   x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{2\pm\sqrt{12}}{2}=\frac{2\pm2\sqrt{3}}{2}=1\pm\sqrt{3}.
   $$

Thus, the equation has two distinct real roots: $$x=1+\sqrt{3}$$ and $$x=1-\sqrt{3}$$.
The equation $$ -x^{2}+2x=-2 $$ has two distinct real roots: $$ x=1+\sqrt{3} $$ and $$ x=1-\sqrt{3} $$.

Discuss the nature of the roots of

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