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 (α+β)3-(γ+δ)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 $(\alpha+\beta)^3-(\gamma+\delta)^3$ is equal to

Complex Numbers and Quadratic Equations

Solution:

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