Question

5. Find the quadratic equation with root alpha and beta,given that alpha-beta=2 and alpha^2-beta^2=3.​

Answer 1

`x^(2)-(3)/(2) x-(7)/(16) =0`

Explanation:

`\begin{cases}\alpha -\beta =2\\ \alpha^(2) -\beta^(2) =3\end{cases}`

`\Longleftrightarrow \begin{cases}\alpha -\beta =2&\\ \left( \alpha +\beta \right) \left( \alpha -\beta \right) =3&\end{cases}`

`\Longleftrightarrow \begin{cases}\alpha -\beta =2&\\ \alpha +\beta =(3)/(2) &\end{cases}`

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 :

`\left( x+(1)/(4) \right) \left( x-(7)/(4) \right) =0`

`\Longrightarrow x^(2)-(3)/(2) x-(7)/(16) =0`