Step-by-step solution doesn't address two solutions given by Reduce command

Questions
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1

Reduce[(5 - 5 x + x^2)^(20 - 9 x + x^2) == 1, x, Integers] gives the full solution set of x={1,2,3,4,5} but the step-by-step solutions (see attached) only show how to find x=1, 4 & 5. The two “missing” solutions are the ones where the base is -1 and, while the expression is undefined (since negative) for almost all values between the roots of x^2 - 5x + 5 =0 approx 1.382 & 3.618 it is defined for x = 2 and x = 3 when the base takes the value -1.

Any thoughts on why the step-by-step solutions don’t deal with the case when the base = -1

Thanks,

Michael

Step by step solution.pdf (197 KB)

2

In addition:
Solve[(x^2 - 5 x + 5)^(x^2 - 9 x + 20) == 1, x]
gives only x=1, 4 & 5 as solutions!

3

Solve[(x^2 - 5 x + 5)^(x^2 - 9 x + 20) == 1, x] generates the message

Solve::ifun: Inverse functions are being used by Solve, so some
solutions may not be found; use Reduce for complete solution
information.

4

Thanks Rohit. That’s very useful.

Any thoughts on why the step-by-step solution doesn’t address the other two solutions?

Yours,

Michael

5

Perhaps the programmer who taught W|A how to solve equations step-by-step had in mind only the real case and did not contemplate that Log in certain situations is a multivalued “function”. Step 2 in the complex domain is wrong: the logarithm of the right-hand side is not zero, but rather 2n*I*Pi.

6

Thanks Gianluca.

7

Hi Michael,

We do not currently support this in our step-by-step solutions, but it is a new feature that we are experimenting with in the Wolfram Alpha Notebooks. For more information on this topic please see the following quick help page:

8

Thanks Elif. When you say “We do not currently support this in our step-by-step solutions” what exactly is “this” referring to?

Thanks,

Michael

9

Hi Michael,
I’m sorry for the confusion. Wolfram|Alpha does not currently support complex numbers within our step-by-step solutions. We are looking into supporting this feature.

Best wishes,

Elif

10

Thanks Elif.

The interesting thing here is that complex numbers aren’t necessarily involved. The two solutions missing from the step-by-step can be found by setting the base euqal to -1, finding the roots and checking that the power is even.

I suppose that in a programming context that’s a bit too much in the realm of “by inspection” . . .

Yours,

Michael