1. **State the problem:** Find all solutions to the equation $x^2 - 8x = -16$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 8x + 16 = 0$$ 3. **Recognize the form:** The equation is a quadratic in standard form $ax^2 + bx + c = 0$ with $a=1$, $b=-8$, and $c=16$. 4. **Check for perfect square:** Notice that $16 = 4^2$ and $-8x = 2 \times 4 \times (-x)$, so the quadratic can be factored as: $$ (x - 4)^2 = 0 $$ 5. **Solve for $x$:** Since the square of $(x-4)$ is zero, the only solution is: $$ x - 4 = 0 \implies x = 4 $$ 6. **Conclusion:** The equation has one unique solution, $x=4$.