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Problem of the Week

Problem E and Solution

What are the Possibilities?


You may be surprised to learn that the equation \((x^2 - 5x + 5)^{x^2+4x-60} = 1\) has five solutions.
Determine all five values of \(x\) that satisfy the equation.


Let’s consider the ways that an expression of the form \(a^b\) can be 1:

Therefore, the values of \(x\) that satisfy \((x^2 - 5x + 5)^{x^2+4x-60} = 1\) are
\(x = -10\), \(x = 1\), \(x = 2\), \(x=4\) and \(x = 6\). There are five values of \(x\) which satisfy the equation.