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polynomial equation

asked 2016-01-13 09:00:32 -0600

Iman Bhamani gravatar image

updated 2016-01-13 10:46:23 -0600

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)
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answered 2016-01-14 04:43:02 -0600

ndomes gravatar image
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)
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answered 2016-01-14 04:36:28 -0600

B r u n o gravatar image

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)
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Asked: 2016-01-13 09:00:32 -0600

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Last updated: Jan 14 '16