An algebra problem by Yeldo Pailo

Algebra Level 3

\(\alpha\) and \(\beta\) are the roots of the quadratic equation \(\lambda ({ x }^{ 2 }-x)+x+5=0\). If \({ \lambda }_{ 1 }\) and \({ \lambda }_{ 2 }\) are the two values of \(\lambda\) for which the roots \(\alpha\) and \(\beta\) are connected by the relation \(\frac { \alpha }{ \beta } +\frac { \beta }{ \alpha } =4\), then find the value of \(\frac { { \lambda }_{ 1 } }{ { \lambda }_{ 2 } } +\frac { { { \lambda } }_{ 2 } }{ { \lambda }_{ 1 } } \).

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