MaplePrimes - answers and comments on Question, Solving equation with radicals

grrr ...

Axel Vogt — Sat, 05 Jan 2013 02:51:25 Z

I say 'grrr', because Maple is weak on that: try to use 'solve' and you see what I mean. That answers means 'any x', which is not correct (I think it follows by sucessive resolving the square roots by squaring).

v:=sqrt(x+3-4*sqrt(x-1))+sqrt(x+8-6*sqrt(x-1)) - 1
is zero for any x between x=5 and x=10 I would say.

Edited: Here is a sketchy way translated from a similar task

subs(x=p+1, v);
expand(%); 
eval(%, p=r^2); simplify(%) assuming 0<r;
convert(%, piecewise,r);
eval(%, r=sqrt(p));
convert(%, piecewise, p);
subs(p=x-1, %);
w:=% assuming 0 <= x-1;

                   {              1/2
                   { 4 - 2 (x - 1)              x <= 5
                   {
              w := {         0                 x <= 10
                   {
                   {               1/2
                   { -6 + 2 (x - 1)           0 < x - 10

Solution

Kitonum — Sat, 05 Jan 2013 03:39:10 Z

Solution by substitution x-1=t^2 :

eq1:=sqrt(x+3-4*sqrt(x-1))+sqrt(x+8-6*sqrt(x-1))=1:

eq2:=x-1=t^2:

simplify(algsubs(eq2, eq1)) assuming t>=0;

sol:=solve(%);

L:=convert(sol, list);

x:=unapply(solve(eq2, x), t);

RealRange(x(L[1]), x(L[2])); # All roots of the initial equation

Of course, after the equation is solved with respect to t , it is easier to solve it by hand for x , using the equality x=1+t^2

With Mathematica

toandhsp — Sat, 05 Jan 2013 08:38:44 Z

I tried solve it with Mathematica

Reduce[Sqrt[x + 3 - 4*Sqrt[x - 1]] + Sqrt[x + 8 - 6*Sqrt[x - 1]] == 
 1, x, Reals]

Thank's for your answers,guys!

VolMike — Sat, 05 Jan 2013 12:40:59 Z

Thanks for your answers,guys! In fact, I was cunning while posting the problem :) I knew the exact solution, it could be easly found by Mathematica (thanks to toandhsp ) or WolframAlpha or it could be also represented as piecewise function( thanks to Axel Vogt ) or it could be plotted (thanks to Preben Alsholm ). All those approaches could give the answer immediately,but the main idea of the question was to find the way of solving those type of equations without any preliminary preparations like substitutions or plotting or so on. Just using solve,isolve or something (internal to Maple) else (look at Mma Reduce as example).Kitonum, am I right and you are the member of russian forum.exponenta.ru?

difficult

Axel Vogt — Sun, 06 Jan 2013 01:45:50 Z

solve(0 = 2-2*(-31/365*x+1)^(1/2)-2*(2-2*(-31/365*x+1)^(1/2)-31/365*x)^(1/2), x) is
something that Wolfram Alpha answers by 0 <= x <= 365/31, which is ~ 11,77.

However for x=12 the command 'simplify' shows that it is zero as well.

I think for that example over the _Reals_ the solution is

piecewise(x < 0, 4-4/365*(-11315*x+133225)^(1/2), 0<=x, 0) ;

plots

Preben Alsholm — Sat, 05 Jan 2013 04:33:23 Z

Playing a little with the two branches of the square root:

restart;
eq:=sqrt(x+3-4*sqrt(x-1))+sqrt(x+8-6*sqrt(x-1)) = 1;
plot([op(eq)],x=1..15,0..2,thickness=2); #Perfectly fine
R:=convert(lhs(eq),RootOf);
plot([R,1],x=1..15,0..2,thickness=2); #Still perfectly fine
subs((index=1)=NULL,R); #Now forgetting about the branch (index)
L:=[allvalues(%)];
plots:-display(seq(plot(L[k],x=-5..15),k=1..nops(L)),insequence=true);

#One of these (number 7 for me) is constantly 1.