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Routine exercises often give students the impression that it is easy to spot the appropriate technique to solve problems in mathematics. The equations in this problem require students to make connections between different areas of mathematics, namely quadratic equations and indices, and to use what they know to solve a problem which at first glance looks much harder than it is.
Display the equation:
(x^2 - 5x + 5)^{(x^2 - 11x + 30)} = 1
Then share the next equation: (x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1
Once students have found all solutions, they may wish to explore other equations in the same family, and there are also some suggestions for Mega Quadratic Equations which have fewer than six solutions.
Do you know any solutions to the equation a^b=1?
How can each solution to a^b=1 help you to find solutions to the equations?
Students may be put off by the apparent complexity of this task, so reassure them that all they need to be able to do is solve quadratic equations and know their index laws! The hint "Do you know any solutions to the equation a^b=1?" may help them to find a fruitful line of enquiry.
Invite students to plot the equations using graphing software such as Desmos. There are some interesting features to explain; in particular, not all six solutions show up graphically without resorting to complex number plots!
The picture below shows the first equation, and also graphs the base of the equation (y=x^2 - 5x - 5). From this it can be seen that Desmos has difficulties graphing y=(x^2 - 5x + 5)^{(x^2 - 11x + 30)} when the base is negative.
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}.