1. Tardigrade
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  4. Consider a biquadratic equation 81 x4+216 x3+216 x2+96 x-65=0 whose roots are α, β, γ, δ. Given α, β are real roots and γ, δ are imaginary roots. The value of γ3+δ3-(α+β)3 is equal to

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Q. Consider a biquadratic equation $81 x^4+216 x^3+216 x^2+96 x-65=0$ whose roots are $\alpha, \beta, \gamma, \delta$. Given $\alpha, \beta$ are real roots and $\gamma, \delta$ are imaginary roots.
The value of $\gamma^3+\delta^3-(\alpha+\beta)^3$ is equal to

Complex Numbers and Quadratic Equations

Solution:

$81 x^4+216 x^3+216 x^2+96 x-65=0$
$\left(3 x x^4+4 C_1(3 x)^3(2)+4 C_2(3 x)^2(2)^2+4 C_3(3 x)(2)^3+(2)^4=81\right.$
$(3 x+2)^4=81$
$(3 x+2)^2=9 \text { or }(3 x+2)^2=-9$
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$\gamma^3+\delta^3-(\alpha+\beta)^3$
$=(\gamma+\delta)\left(\gamma^2+\delta^2-\gamma \delta\right)-(\alpha+\beta)^3 $
$=\frac{-4}{3}\left(\left(\frac{-4}{3}\right)^2-3 \gamma \delta\right)-\left(\frac{-4}{3}\right)^3$
$=\frac{4}{3} \cdot 3 \cdot \frac{13}{9}=\frac{52}{9}$