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solving radical equations with parameters

asked 11 years ago

anonymous user

Anonymous

I would like to find the solutions y to this type of equations: (1+x(1+x)24y)2=z with conditions on x,y,z (like 0<y<x18 and 0<z<x2).

Using solve with the option to_poly_solve:

sage: solve((1+x - sqrt((1+x)^2-4*y))^2 == z, y, to_poly_solve=True)                                                 
[y == 1/2*x^2 - 1/2*(x + 1)*sqrt(x^2 + 2*x - 4*y + 1) + x - 1/4*z + 1/2]

does not seem to work because y appears on the right side of the solution. I expect to find a solution like y=14((1+x)2(1+xz)2).

I also tried the same after specifying the conditions with assume(), without success.

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answered 11 years ago

ndomes gravatar image

I tinkered with your equation and finally had something that looks like a solution. But this is not the way I expect solving equations with sage.

var('x y z')
expr = (1+x - sqrt((1+x)^2-4*y))^2
eqn1 = expr.expand().simplify_full() == z
show(eqn1)

eqn2 = eqn1.subtract_from_both_sides(2*x^2 +4*x -4*y +2)
eqn3 = eqn2.lhs()^2 == eqn2.rhs()^2
show(eqn3)

eqn4 = eqn3.expand()
eqn5 = eqn4.subtract_from_both_sides(eqn4.lhs())
show(eqn5)
sol = solve(eqn5,y)
for s in sol: show(s)
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Asked: 11 years ago

Seen: 683 times

Last updated: Jun 26 '13