System of equations from coefficients of series

Question

I have two functions $F_1(a,b,x)$ and $F_2(a,b,x)$ I expand these functions about the point $x=0$

  Series[F1,{x,0,2}]
  Series[F2,{x,0,2}]

the coefficients are depended on $a$ and $b$ variables. I want to create the system of equations where all coefficients are equal to zero.

$$F_1(a,b,x)=A_0(a,b)+A_1(a,b)x+A_2(a,b)x^2$$ $$F_2(a,b,x)=B_0(a,b)+B_1(a,b)x+B_2(a,b)x^2$$

so the system of the equations is

$A_0(a,b)=0, \ A_1(a,b)=0, \ A_2(a,b)=0, B_0(a,b)=0, \ B_1(a,b)=0, \ B_2(a,b)=0$

I know that CoefficientList[Series[F1,{x,0,2}],{x}] returns coefficients of the expansion, but how can I use them to create equations? And how can I pass them to Solve[] function?

Answer 1

As suggested J.M. above I use Thread[] function which converts list to list of equations, for example Thread[{a,b,c}==0] = {a=0,b=0,c=0}. So I have two lists

coeff1 = CoefficientList[Series[F1,{x,0,2}],{x}]
coeff2 = CoefficientList[Series[F1,{x,0,2}],{x}]

I join them using Join[] function

coeffs = Join[coeff1,coeff2];

and then I solve the system of equations

 Solve[Thread[coeffs == 0], {a,b}]