Question

Let $\alpha +\beta =3$ and ${\alpha }^3+{\beta }^3=7$, then find the equation of the quadratic equation whose roots are $\alpha ,\beta$.

Answer 1

Given,

$\alpha +\beta =3$......(1)

And

${\alpha }^3+{\beta }^3=7$

or, $(\alpha +\beta {)}^3-3\alpha .\beta (\alpha +\beta )=7$

or, $3^3-9\alpha .\beta =7$ [ Using (1)]

or, $9\alpha .\beta =2$

or, $\alpha .\beta =\frac{20}{9}$......(2).

Now the quadratic equation whose roots are $\alpha ,\beta$ is

$x^2-(\alpha +\beta )x+\alpha .\beta =0$

or, $x^2-3x+\frac{20}{9}=0$

or, $9x^2-27x+20=0$. [Using (1) and (2)]