How can I solve it by sage?
Sage: solve(-6-4*y-x-(1+y)*x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)==0, (2*(4+2*y+x))*(1+y)-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)-(2+y)*(-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2))==0, x,y)
It is not clear in your question on which field or ring you want to solve your equation. Though the following sequence of computations seems to indicate that your system is verified by any value for x and y:
sage: var('x,y')
(x, y)
sage: p = -6-4*y-x-(1+y)*x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)
sage: q = (2*(4+2*y+x))*(1+y)-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)-(2+y)*(-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2))
sage: s = solve(p==0,x,y); s
([x == -(4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2)],
[1])
sage: x0 = s[0][0].rhs()
sage: q.subs(x=x0)
-((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) + sqrt(-4*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1)*((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) - 2) + ((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2)^2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2)*(y + 2) + 2*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1) + (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) + sqrt(-4*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1)*((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) - 2) + ((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4 ...
var('x y')
eq1 = -6-4*y-x-(1+y)*x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)==0
eq2 = (2*(4+2*y+x))*(1+y)-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)-(2+y)*(-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2))==0
show(eq1)
eq11 = eq1 + 6 +x*(y+1) + x + 4*y
eq12 =(eq11^2).expand()
show(eq12)
eq13 = eq12 - eq12.lhs()
show(eq13)
print '------------'
show(eq2)
eq21 = eq2.simplify_full()
show(eq21)
eq22 = eq21 - 6 -2*x -(x+4)*y^2 -(3*x+10)*y
show(eq22)
eq23 = (eq22^2).expand()
show(eq23)
eq24 = eq23 - eq23.lhs()
show(eq24)
Asked: 2016-01-13 16:00:32 +0100
Seen: 589 times
Last updated: Jan 14 '16
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