# How to solve a complicated equation?

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Cola on 7 Jul 2021
Edited: David Goodmanson on 16 Jul 2021
There is a Equation G. How to obtain the values of α and β when G=0？
G=-(-Omega^3*tau + (alpha + beta)*Omega)^2 - (Omega^2 - alpha*f)^2.
alpha = Omega^2*cos(Omega*tau)/f;
beta = Omega*(f*sin(Omega*tau) - Omega*cos(Omega*tau))/f.
CAN anyone help me with this issue??? Thanks!!!
Star Strider on 16 Jul 2021
My pleasure!

David Goodmanson on 16 Jul 2021
Edited: David Goodmanson on 16 Jul 2021
Hi Cola,
Since there is one equation and two unknowns, it must be possible to define, say, beta in terms of alpha, where alpha can be anything. For G = 0 we have
(-Om^3*t + (a+b)*Om)^2 = -(Om^2 - a*f)^2
so
-Om^3*t + (a+b)*Om = +-*i*(Om^2 - a*f)
where there are obvious notational substitutions for Omega, tau, alpha, beta, and the +- choice gives two different solutions. Solving for b,
b = (1/Om)*( Om^3*t -a*Om +-i*(Om^2 - a*f) )
where 'a' can be anything. Solving instead for a (this does not give a different family of solutions, rather the same ones expressed differently) gives
a = (Om^3*t +-i*Om^2 -b*Om)/(Om +-i*f)
Here the sign in the denominator (+ or -) has to match the sign in the denominator, and b can be anything. The choice b = 0 gives the solutions from Star Strider.
Cola on 16 Jul 2021
I really thank you and sincerely wish you all the best.