Question

Determinati numerele reale x si y care au suma egala cu 2 si produsul egal cu -15.

Answer 1

$\left \{ {{x+y=2} \atop {xy=-15}} \right. \Leftrightarrow \left \{ {{y=2-x} \atop {xy=-15}} \right. \Leftrightarrow \left \{ {{y=2-x} \atop {x(2-x)=-15}} \right. \Leftrightarrow \left \{ {{y=2-x} \atop {2x- x^{2} =-15}} \right. \Leftrightarrow \left \{ {{y=2-x} \atop {- x^{2} +2x+15=0}} \right. .$

$- x^{2} +2x+15=0 \\ \Delta= b^{2}-4ac=4-4*(-1)*15=64. \\ \\ x_{1,2}= \frac{-b \pm \sqrt{\Delta} }{2a}= \frac{-2 \pm8}{-2}= \left \{ {{5} \atop {-3}} \right. . \\ x=5 \Rightarrow y=2-5=-3. \\ x=-3 \Rightarrow y=2-(-3)=5. \\ \\ \underline{Solutie}: \boxed{(x,y) \in \{ (-3;5);(5;-3)\}}.$