If (x + y + z) = 0, then what is the value of $$\frac{{3{y^2} + {x^2} + {z^2}}}{{2{y^2} - xz}}?$$

Solution (By Examveda Team)

$$\eqalign{ & x + y + z = 0,\,\frac{{3{y^2} + {x^2} + {z^2}}}{{2{y^2} - xz}} = ? \cr & {\text{Put }}x = 1,\,y = 1,\,z = - 2 \cr & \Rightarrow \frac{{3{{\left( y \right)}^2} + {x^2} + {z^2}}}{{2{y^2} - xz}} \cr & \Rightarrow \frac{{3{{\left( 1 \right)}^2} + {{\left( 1 \right)}^2} + {{\left( { - 2} \right)}^2}}}{{2{{\left( 1 \right)}^2} - \left( {1 \times \left( { - 2} \right)} \right)}} \cr & \Rightarrow \frac{{4 + 4}}{{2 + 2}} \cr & \Rightarrow \frac{8}{4} \cr & \Rightarrow 2 \cr} $$