Answer:
[tex]x^{2}-\frac{3}{2} x-\frac{7}{16} =0[/tex]
Step-by-step explanation:
[tex]\begin{cases}\alpha -\beta =2\\ \alpha^{2} -\beta^{2} =3\end{cases}[/tex]
[tex]\Longleftrightarrow \begin{cases}\alpha -\beta =2&\\ \left( \alpha +\beta \right) \left( \alpha -\beta \right) =3&\end{cases}[/tex]
[tex]\Longleftrightarrow \begin{cases}\alpha -\beta =2&\\ \alpha +\beta =\frac{3}{2} &\end{cases}[/tex]
Then
2α = 2 + 3/2 = 7/2
2β = (3/2) - 2 = -1/2
Then
Then
α = 7/4
β = -1/4
Then
a quadratic equation with root α and β can be :
[tex]\left( x+\frac{1}{4} \right) \left( x-\frac{7}{4} \right) =0[/tex]
[tex]\Longrightarrow x^{2}-\frac{3}{2} x-\frac{7}{16} =0[/tex]
