Rid polynomial of square-root



David Billinghurst wrote:
> Russell Shaw wrote:
>> Hi,
>> Starting with:
>>
>>   x = sqrt(x^2 + 1) - a
>>
>> How do i get it into the form:
>>
>>   x^2(b^2 - 1) + 2*a*b*x + a^2 - 1 = 0
>>
>>
>> Also, i tried to solve for x:
>>
>>                     2
>> (%o1)     x = sqrt(x  + 1) + a
>>
>> (%i2) solve(%, x);
>>
>>
>> It just gives:
>>                     2
>> (%o2)    [x = sqrt(x  + 1) + a]
>>
>> I've read all the manuals.
>>
> You can solve it "by hand" using maxima.  Where does where b come into it?

Hi David.B,

It should have been: x = (sqrt(x^2 + 1) - a)/b

Too many hours staring at manuals and stuff.

> (%i1) x=sqrt(x^2+1)-a;
>                                       2
> (%o1)                        x = sqrt(x  + 1) - a
> (%i2) %+a;
>                                           2
> (%o2)                        x + a = sqrt(x  + 1)
> (%i3) lhs(%)^2=rhs(%)^2;
>                                      2    2
> (%o3)                          (x + a)  = x  + 1
> (%i4) expand(%);
>                            2            2    2
> (%o4)                      x  + 2 a x + a  = x  + 1
> (%i5) lhs(%)-rhs(%)=0;
>                                       2
> (%o5)                         2 a x + a  - 1 = 0
> (%i6) solve(%,x);
>                                        2
>                                       a  - 1
> (%o6)                           [x = - ------]
>                                        2 a

I had thought of manually solving it like this, but was wondering if
there was a more automated way of solving for x, because the example
is a simpler form of a larger messier equation i was working on.

I was also unsure how to manipulate equations manually in maxima
that well having found Maxima a few days ago, but your example
makes it clear.