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

Posted 3 years ago
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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

POSTED BY: Michael Meagher
9 Replies

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

POSTED BY: Michael Meagher
Posted 3 years ago

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.

POSTED BY: Rohit Namjoshi

Thanks Rohit. That's very useful.

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

Yours,

Michael

POSTED BY: Michael Meagher

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". StepĀ 2 in the complex domain is wrong: the logarithm of the right-hand side is not zero, but rather 2n*I*Pi.

POSTED BY: Gianluca Gorni

Thanks Gianluca.

POSTED BY: Michael Meagher

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: https://www.wolfram.com/wolfram-alpha-notebook-edition/quick-help/

POSTED BY: Elif Basar

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

POSTED BY: Michael Meagher

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

POSTED BY: Elif Basar

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

POSTED BY: Michael Meagher
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