Given unknown variables $x_1, x_2, x_3$
And given known constants $c_1, c_2, c_3, c_4, c_5, c_6, c_7, c_8, c_9, d_1, d_2, d_3$
And three equations of the form:
$c_1x_1^2 + c_2x_2^2 + c_3x_1x_2 = d_1$
$c_4x_1^2 + c_5x_3^2 + c_6x_1x_3 = d_2$
$c_7x_2^2 + c_8x_3^2 + c_9x_2x_3 = d_3$
Does a closed form solution exist? If so, how can it be derived? I am already aware of several iterative methods to solve this, but was hoping to be able to avoid those. Thanks!