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These are questions asked by minhthien2016

The equation sin(9*x-(1/3)*Pi) = sin(7*x-(1/3)*Pi) can be solved easy by hand with solutions k*Pi and -Pi/48 + K*Pi/8. With Maple, I tried 
solve({sin(9*x-(1/3)*Pi) = sin(7*x-(1/3)*Pi)}, x, explicit, allsolutions)

I don't get the above solutions. How can I get these solutions?

Based on the equation at here https://www.mapleprimes.com/questions/209660-Problem-With-RealDomainsolve

I tried solve the equation (x-1)*sqrt(x^2 - 4)=0 in Real domain. My code
RealDomain:-solve((x-1)*sqrt(x^2-4) = 0, x);

I got there solutions are 1, 2, -2.  I think, If we solve the given in RealDomain, we only get two solutions -2 and 2.

My question is: How many solutions are there in the equation (x-1)*sqrt(x^2 - 4)=0 by RealDomain:-solve?

Two pictures by using Mathematica.

I am trying to find all real solutions of the system equations 
sol := solve(And(g'(x)=0,g''(x)<>0),x)   assuming real;
I tried

restart; fprime := x-> x^6-(3/2)*x^5+2*x^4+(5/2)*x^3-7*x^2+2:
f := unapply(simplify(int(fprime(x), x)), x):
g := unapply(expand(f(x^2+2*x)), x):
sol := solve(And(g'(x)=0,g''(x)<>0),x)   assuming real;

I don't get only real solutions. How can I get only real solutions?

How to get the only solution x = (1/6)*sqrt(114)*a of this equation?
u := `<,>`(-(1/4)*a, -(1/12)*sqrt(3)*a, -x);
v := `<,>`(-(1/2)*a, (1/6)*sqrt(3)*a, (1/2)*x);
`assuming`([solve(Angle(u, v) = arccos(2*sqrt(26)*(1/13)), x)], [And(a > 0, x > 0)])

I have the sequence f(n) satify the conditions

f(n) = 4*f(n-1)-2*f(n-2), f(1) = 1, f(2) = 28.

I find the formula of f(n)

rsolve({f(1) = 1, f(2) = 28, f(n) = 4*f(n-1)-2*f(n-2)}, {f})

I got

f(n) = (-25*sqrt(2)*(1/4)-6)*(2-sqrt(2))^n+(25*sqrt(2)*(1/4)-6)*(2+sqrt(2))^n

Is there an integeral number k satisfy the equation f(n) = k^2, (k <> 0) ?

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