\( \left\{ \begin{array} { l } { 2 x _ { 1 } x _ { 2 } - 2 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 0 } = 0 } \\ { x _ { 1 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + 2 x _ { 2 } x _ { 0 } = 0 } \\ { - x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } = 0 } \end{array} \right. \)

To solve this system of equations, start by analyzing the third equation: \( -x_1^2 + x_2^2 = 0 \). This simplifies to \( x_2^2 = x_1^2 \), which means \( x_2 = x_1 \) or \( x_2 = -x_1 \). You can substitute \( x_2 \) back into the first two equations to determine more about \( x_1 \) and \( x_0 \). You might find it interesting that systems of equations like this one can model real-world scenarios, such as equilibrium in physical systems or optimization problems in economics. For example, imagine having two businesses that need to optimize their costs while managing shared resources — solving these kinds of equations can help in practical decision-making. Don't forget to check for common mistakes like overlooking the possibilities of negative values or missing out on simplifying the equations further. When handling multiple variables, it's easy to misinterpret relationships or lose track of constraints. It's important to be methodical and double-check each step along the way to avoid getting tangled in the calculations!