Question
Let $a,b \in C$. Let $\alpha,\beta$ be the roots of the equation $x^{2} + ax + b = 0$. If $\beta - \alpha = \sqrt{11}$ and $\beta^{2} - \alpha^{2} = 3i\sqrt{11}$, then $\left( \beta^{3} - \alpha^{3} \right)^{2}$ is equal to:
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Solution
Sol. [Key Concept: Factor $\beta^{2} - \alpha^{2} = (\beta + \alpha)(\beta - \alpha)$ to get $\beta + \alpha$; then $(\beta^{3} - \alpha^{3})^{2} = (\beta - \alpha)^{2}((\beta + \alpha)^{2} - \alpha\beta)^{2}$.]
$$\beta - \alpha = \sqrt{11} $$and $\beta^{2} - \alpha^{2} = 3i\sqrt{11}$
$$\Rightarrow (\beta + \alpha)(\beta - \alpha) = 3i\sqrt{11}$$
$$\begin{array}{r} \Rightarrow \beta + \alpha = 3i\#(2) \end{array}$$
from (1) and (2)
$${\beta = \frac{\sqrt{11} + 3i}{2},\ \alpha = \frac{3i - \sqrt{11}}{2} }{\Rightarrow \alpha\beta = \frac{- 9 - 11}{4} \Rightarrow \alpha\beta = - 5 }$$Now $\left( \beta^{3} - \alpha^{3} \right)^{2} = (\beta - \alpha)^{2}\left( \beta^{2} + \alpha^{2} + \alpha\beta \right)^{2}$
$${= (\beta - \alpha)^{2}\left( (\beta + \alpha)^{2} - \alpha\beta \right)^{2} }{= (11)( - 9 + 5)^{2} = 176 }$$
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