# If $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$, then find the quadratic equation whose roots are $\alpha$ and $\beta$.

Given: $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$.

To do: To find the quadratic equation whose roots are $\alpha$ and $\beta$.

Solution:

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 $x^2-(-2)x+(-8)=0$

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

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