Why does sage solve() return another equation(when it is clearly simple and solvable)

Question

I am trying to solve a simple equation on sage with the given data:

a=var("a")
x=vector([1,1])
y=vector([2,3])
A=matrix(2,2,[a,-1,-1,1])

I want to solve the equation:

f= ((x*A*y)/((sqrt(x*A*x)*sqrt(y*A*y))))==1/2 

which returns :

(2sqrt(a - 1)/sqrt(4a - 3) == (1/2)

solving f for a returns:

solve(f,a)

[sqrt(a - 1) == 1/4sqrt(4a - 3)]

while this equation is easily solvable by hand. Is there something I'm missing or doing wrong?

Answer 1

Sage is not implementing every possible way to search for a solution. In my way of using it, the combination of human (structural) insight and the computational power of the machine always works best.

Why do we have the half-way answer? In our case f is an equation,

sage: f
2*sqrt(a - 1)/sqrt(4*a - 3) == (1/2)

which is not really an algebraic equation. (Square roots appear in there.) The documentation of solve, ask for it via solve? or ?solve or... , mentions this algebraic character wanted for the involved equations / expressions. Sage may try on its own to get an algebraic version, but this is too much a voluntary step, depending on the typed solve command line.

So it is a normal fact to get an "echo" of the equation.

Note however, that involving an other solver helps (in this case):

sage: solve(f, a, algorithm='sympy')
[a == (13/12)]

Answer 2

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