Algebra · Systems of equations · Perfect squares · Quadratic equations · Vieta relations
Find all real numbers \(x\), \(y\), \(z\) that satisfy the system
\[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2009, 1. letnik, category A, problem 1. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source