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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