I need to solve this nonlinear system of equations. Coefficients a,b,c,d are given. Since it is a system of 4 eq. and 4 unknowns $x_{i}$ it should have a solution but unfortunately there are the squares... Is there any way to solve this in relatively elegant way? I tried but only get a mess...
$ax_{1}^2 + 2bx_{1}x_{3} + cx_{3}^2-a=0$
$ax_{1}x_{2} + bx_{1}x_{4} + bx_{2}x_{3} + cx_{3}x_{4} - b=0$
$ax_{2}^2+2bx_{2}x_{4} +cx_{4}^2 -c= 0$
$x_{1}x_{4}-x_{2}x_{3}=1$