I am trying to solve this equation in sage √−4z2+2√−4z2+1−1=0. But when I try the code
var('z')
eq = sqrt(-4*z^2 + 2*sqrt(-4*z^2 + 1) - 1) == 0
solve(eq,z)I get
[z == -1/2*sqrt(2*sqrt(-4*z^2 + 1) - 1), z == 1/2*sqrt(2*sqrt(-4*z^2 + 1) - 1)]Is there any way to actually solve it in sage?
To better specify the question, one would need to say if we want to solve for z in Z, Q, R, C, or other.
Here we examine how to solve over C.
Define z as a symbolic variable, and define the equation (as you did).
sage: z = SR.var('z')
sage: eq_a = sqrt(-4*z^2 + 2*sqrt(-4*z^2 + 1) - 1) == 0
sage: solve(eq_a, z)
[z == -1/2*sqrt(2*sqrt(-4*z^2 + 1) - 1), z == 1/2*sqrt(2*sqrt(-4*z^2 + 1) - 1)]First notice that √expression=0 is equivalent to expression=0. This gets you rid of one of the square roots.
sage: eq_b = eq_a^2
sage: eq_b
-4*z^2 + 2*sqrt(-4*z^2 + 1) - 1 == 0Then isolate the other square root.
sage: eq_c = eq_b + 4*z^2 + 1
sage: eq_c
2*sqrt(-4*z^2 + 1) == 4*z^2 + 1Remove it by squaring. Note that the new equation is implied by the initial one, but no longer equivalent to it.
sage: eq_d = eq_c^2
sage: eq_d
-16*z^2 + 4 == (4*z^2 + 1)^2Solve the new equation.
sage: sols_d = solve(eq_d, z)
sage: sols_d
[z == -1/2*sqrt(2*sqrt(3) - 3), z == 1/2*sqrt(2*sqrt(3) - 3), z == -1/2*sqrt(-2*sqrt(3) - 3), z == 1/2*sqrt(-2*sqrt(3) - 3)]Check which of the solutions are solutions of the initial equation.
sage: [bool(eq_a.subs(s)) for s in sols_d]
[True, True, False, False]Define the solutions of the original equation:
sage: sols_a = [s for s in sols_d if bool(eq_a.subs(s))]
sage: sols_a
[z == -1/2*sqrt(2*sqrt(3) - 3), z == 1/2*sqrt(2*sqrt(3) - 3)]Asked: 7 years ago
Seen: 1,336 times
Last updated: Dec 31 '17
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To better specify the question, do you want to solve for z in Z, Q, R, C, other?
One thing to notice is that √expression=0 is equivalent to expression=0. This gets you rid of one of the square roots.
This has two solutions in C, ±√2√3−3/2.
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