I think sage solving is based on maxima, and I've noticed that Maxima has
always had this problem with non-linear equations.
If you set it up as follows:
eq = x == sqrt(1 + x)
solns = solve(eq^2,x); solns
You will get:
[x == -1/2*sqrt(5) + 1/2, x == 1/2*sqrt(5) + 1/2]
Of course, then you have to do the following:
for i in solns:
n(eq.subs(i).lhs()) == n(eq.subs(i).rhs())
Which returns:
False
True
With a bit of playing around, you could turn this into a lambda and a filter
to return the correct solutions. (I'm sure people on this group can take
this approach and give you something more elegant.)
I tend to think of these situations as, the CAS isn't there to do the entire
job for you, but rather support you but doing the basic heavy lifting for
you.
In circumstances like this, you have to guide the CAS to the correct
solution.
Joal Heagney
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