Question

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

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

Answer 1

Answer: (D)

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