Question

If $\alpha +\beta =-2$ and ${\alpha }^3+{\beta }^3=-56$, then the quadratic equation whose roots are $\alpha$ and $\beta$ 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 $x^2+2x-8=0$

Given that, $\alpha +\beta =-2$ and ${\alpha }^3+{\beta }^3=-56$
$\Rightarrow (\alpha +\beta )({\alpha }^2+{\beta }^2-\alpha \beta )=-56$
$\Rightarrow {\alpha }^2+{\beta }^2-\alpha \beta =28$
Now, ${(\alpha +\beta )}^2={(-2)}^2$
$\Rightarrow {\alpha }^2+{\beta }^2+2\alpha \beta =4$
$\Rightarrow 28+3\alpha \beta =4$
$\Rightarrow \alpha \beta =-8$
$\therefore$ Required equation is
$x^2-(-2)x+(-8)=0$
$\Rightarrow x^2+2x-8=0$