Question

If the sum of the roots of an equation is 2 and the sum of their cubes is 98, then find the quadratic equation.

Answer 1

$\Rightarrow$ Let $\alpha$ and $\beta$ are roots of the required equation.

$\Rightarrow$ $\alpha +\beta =2$ [ Given ] ---- ( 1 )

$\Rightarrow$ ${\alpha }^3+{\beta }^3=98$

$\Rightarrow$ $(\alpha +\beta )({\alpha }^2-\alpha \beta +{\beta }^2)=98$

$\Rightarrow$ $2({\alpha }^2+2\alpha \beta +{\beta }^2-3\alpha \beta )=98$ [ Using ( 1 ) ]

$\Rightarrow$ $\left[\right(\alpha +\beta {)}^2-3\alpha \beta ]=49$

$\Rightarrow$ $(2{)}^2-3\alpha \beta =49$

$\Rightarrow$ $4-3\alpha \beta =49$

$\Rightarrow$ $3\alpha \beta =-45$

$\therefore$ $\alpha \beta =-15$ ----- ( 2 )

New equation,

$\Rightarrow$ $x^2-(\alpha +\beta )x+\left(\alpha \beta \right)=0$

Using ( 1 ) and ( 2 ),

$\Rightarrow$ $x^2-2x-15=0$