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Topic: Is there a general theory of solving equations?
Replies: 6   Last Post: Aug 24, 2013 9:07 AM

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 Ken.Pledger@vuw.ac.nz Posts: 1,332 Registered: 12/3/04
Re: Is there a general theory of solving equations?
Posted: Aug 21, 2013 5:53 PM

Kobu <kobu.selva@gmail.com> wrote:

> Is there a general theory of solving equations?

That depends on what you mean by "solving". An equation such as
cos(x) = x has a solution (as you can see from where the two graphs
cross), but calculating it would require a numerical method of
successive approximation. The more work you did, the more decimal
places you could find.

> .... Then we do some steps.. symbolic mashing of both sides... or the rule "do the
> same thing to both sides" ...

That's fine if you have a simple algebraic equation, but the above
example shows that life isn't always so simple.

> .... proving when
> polynomial equations are solvable via neat formulas (like quadratic formula).
> Basically proving that no such formula exists for degree > 4.

So even for many polynomial equations, there's no escaping numerical
approximations.

> .... 5) Dividing by an expression in x that might be zero at a, b, c, *might* make
> you "lose" solutions a, b, c.

For that reason, it may be clearer to factorise rather than divide.
For example,
if (x - 1)(3x + 2) = (x - 1)(x - 4)
then rearrange the equation as
(x - 1)((3x + 2) - (x - 4)) = 0
so (x - 1)(2x + 6) = 0.
Then infer that at least one of the factors is 0, etc. To me this seems
neater than checking separately that you haven't tried to divide by zero.

> .... What's the general theory?

There isn't any _completely_ general theory. That's why existence
and uniqueness theorems are important. People have proved that
equations of various kinds (including many differential equations) have
at least one solution and/or at most one solution. After that we know
whether it makes sense to start computing numerically.

Ken Pledger.

Date Subject Author
8/21/13 Kobu
8/21/13 JW
8/21/13 Brian Q. Hutchings
8/21/13 Ken.Pledger@vuw.ac.nz
8/23/13 tommy1729_
8/24/13 JW
8/24/13 JW