Question

If $\alpha +\beta$$=-2$ and ${\alpha }^3+{\beta }^3$$=-56$ then the quadratic equation whose roots are $\alpha ,\beta$ is

• A. $x^2+2x-16$$=0$
• B. $x^2+2x-15$$=0$
• C. $x^2+2x-12$$=0$
• D. $x^2+2x-8$$=0$

Answer 1

The correct option is D $x^2+2x-8$$=0$

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

$\Rightarrow$ ${\alpha }^3+{\beta }^3=-56$

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

$\Rightarrow$ $(\alpha +\beta {)}^3={\alpha }^3+{\beta }^3+3\alpha \beta (\alpha +\beta )$

$\Rightarrow$ $(-2{)}^3=-56+3\alpha \beta (-2)$ [ Using ( 1 ) and ( 2 ) ]

$\Rightarrow$ $-8+56=-6\alpha \beta$

$\Rightarrow$ $48=-6\alpha \beta$

$\Rightarrow$ $\alpha \beta =-8$ ----- ( 2 )

The required quadratic equation,

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

Using ( 1 ) and ( 3 ) we get,

$\Rightarrow$ $x^2+2x-8=0$