# Solving a non-linear system of equations with multiple variables

I am trying to solve an matrix equation of the form X^2 + AX + B = 0. After the multiplication of the matrices and after adding the matrices we get the following non-linear system of equations: $a^2+bc+a+c\:=\:7;$ $ab+bd+b+d=-1;$ $ac+cd-a+c=0;$ $d^2+bc-b+d=0$

I've tried factoring, tried to eliminate some of the unknowns but can't find a way to solve this system.

Can somebody help me with it ?

• $$\left(a = \dfrac{21}{10},b = -\dfrac{3}{10},c = \dfrac{7}{10},d = -\dfrac{1}{10}\right),(a = -3,b = 0,c = 1,d = -1),(a = 2,b = 0,c = 1,d = -1),(a = 0,b = 0,c = 7,d = -1),\left(a = -\dfrac{14}{5},b = \dfrac{2}{5},c = \dfrac{7}{5},d = -\dfrac{1}{5}\right),(a = 0,b = 6,c = 1,d = -1)$$ – Raffaele Sep 17 '17 at 14:40

eliminating the variables $$b,c,d$$ we get for the following equation for $a$: $$\left( a+3 \right) \left( a-2 \right) \left( 14+5\,a \right) \left( -21+10\,a \right) =0$$ don't Forget to discuss the Special cases