# polynomial equation

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)

edit retag close merge delete

Sort by ยป oldest newest most voted

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 ...
more
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)

more