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Mega Quadratic Equations printable sheet
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.
Find S_r = 1^r + 2^r + 3^r + ... + n^r where r is any fixed positive integer in terms of S_1, S_2, ... S_{r-1}.