Talk:Extraneous and missing solutions: Difference between revisions

Revision as of 01:56, 19 January 2008

Revision as of 06:30, 19 January 2008

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How about an example where we're seeking strictly real solutions to a quadratic equation that also has "extraneous" imaginary solutions? Alternatively, maybe we create an example where we have an explicit given ___domain where we are only interested in positive solutions - but, there are also negative extraneous solutions. [[User:Tparameter|Tparameter]] ([[User talk:Tparameter|talk]]) 01:56, 19 January 2008 (UTC)

:I agree that the example is rather transparent, but students learning algebra do stumble over such things. We want to keep the example simple. I don't know if there is an "official" definition of solutions being extraneous, but the way I understand the term they are solutions introduced by transforming the equation and not solutions arising by interpreting the equation in an extended ___domain.

:One possible replacement is:

::1/(x²+3x−2) = 1/(x²+x+2).

:Take the inverse of both sides:

::x²+3x−2 = x²+x+2.

:Bring everything to one side:

::2x−4 = 0.

:So x = 2 – which is not a solution.

:Another possible example:

::x²+x+1 = 0.

:Multiply by x:

::x³+x²+x = 0.

:Subtract the original equation:

::x³−1 = 0.

:So x = 1 – which is not a solution.

: --[[User talk:Lambiam|Lambiam]] 06:30, 19 January 2008 (UTC)