Challenge Level

Find all real solutions of the equation

$$(x^2-5x+5)^{(x^2-11x+30)} = 1$$

There are six possible solutions to the equation - did you find all six?

Here are some more questions to think about

1. Find all the solutions to $(x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1$.

How do these solutions compare to the first equation?

2. Can you find a Mega Quadratic Equation with solutions $3, 4, 5, 6, 7, 8$?

How about $4, 5, 6, 7, 8, 9$?...

3. Can you explain why there are only $4$ solutions to $(x^2-5x+5)^{(x^2-4)}=1$?

4. Can you explain why there are only $3$ solutions to $(x^2-6x+10)^{(x^2+x-2)}=1$?

5. Can you find a Mega Quadratic equation with exactly $2$ solutions? $5$ solutions?

With thanks to *Don Steward*, whose ideas formed the basis of this problem.