1. The problem involves solving the equation $$5^x = x^5$$ and analyzing the quadratic equation $$x^2 - nx = 0$$ where $$x = n$$ is a parameter. 2. First, consider the quadratic equation $$x^2 - nx = 0$$. We can factor it as $$x(x - n) = 0$$. 3. According to the zero product property, if $$x(x - n) = 0$$, then either $$x = 0$$ or $$x = n$$. 4. The equation $$5^x = x^5$$ is transcendental and can be analyzed graphically or numerically. It represents points where the exponential function $$5^x$$ intersects the power function $$x^5$$. 5. The graph shows a curve resembling an exponential function with a shaded area under the curve between two points on the x-axis, likely between the roots of the quadratic or solutions to the transcendental equation. 6. To summarize: - The quadratic equation $$x^2 - nx = 0$$ has solutions $$x = 0$$ and $$x = n$$. - The transcendental equation $$5^x = x^5$$ requires numerical methods or graphing to find approximate solutions. Final answers: - Quadratic roots: $$x = 0$$ and $$x = n$$. - Transcendental equation solutions: numerical or graphical approximation needed.