tìm 2 số u và v biết 1, u+v= 11 u . v= 30 2, u – v = 2 u . v = 15 3, u^2 + v^2 = 29 u . v = 10

Đáp án:

$\begin{array}{l}
1)\left\{ \begin{array}{l}
u + v = 11\\
u.v = 30
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
u = 11 – v\\
\left( {11 – v} \right).v = 30
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
u = 11 – v\\
{v^2} – 11v + 30 = 0
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
v = 6\\
u = 5
\end{array} \right.\\
\left\{ \begin{array}{l}
v = 5\\
u = 6
\end{array} \right.
\end{array} \right.\\
2)\left\{ \begin{array}{l}
u – v = 2\\
u.v = 15
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
u = 2 + v\\
\left( {v + 2} \right).v = 15
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
u = 2 + v\\
{v^2} + 2v – 15 = 0
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
u = 2 + v\\
\left[ \begin{array}{l}
v = 3\\
v =  – 5
\end{array} \right.
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
v = 3;u = 5\\
v =  – 5;u =  – 3
\end{array} \right.\\
3)\left\{ \begin{array}{l}
{u^2} + {v^2} = 29\\
u.v = 10 \Rightarrow u = \frac{{10}}{v}
\end{array} \right.\\
 \Rightarrow \frac{{100}}{{{v^2}}} + {v^2} = 29\\
 \Rightarrow {v^4} – 29{v^2} + 100 = 0\\
 \Rightarrow \left( {{v^2} – 4} \right)\left( {{v^2} – 25} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
{v^2} = 4 \Rightarrow {u^2} = 25\\
{v^2} = 25 \Rightarrow {u^2} = 4
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
u = 2;v = 5\\
u =  – 2;v =  – 5\\
u = 5;v = 2\\
u =  – 5;v =  – 2
\end{array} \right.
\end{array}$