Let . Let be the roots of the equation . If and , then is equal to : [2026]

Question

Let $a, b \in C$ . Let $α, β$ be the roots of the equation $x^{2} + a x + b = 0$ . If $β - α = \sqrt{11}$ and $β^{2} - α^{2} = 3 i \sqrt{11}$ , then ${(β^{3} - α^{3})}^{2}$ is equal to : [2026]

Smart Achievers · Accepted Answer

(2)

Given, $x^{2} + a x + b = 0$ , $β - α = \sqrt{11}$ and $β^{2} - α^{2} = 3 i \sqrt{11}$

$\Rightarrow (β + α) (β - α) = 3 i \sqrt{11}$

$\Rightarrow β + α = 3 i$

$\Rightarrow β^{2} + α^{2} + 2 α β = - 9$ ... (i)

$∵ β - α = \sqrt{11}$

$\Rightarrow β^{2} + α^{2} - 2 α β = 11$ ... (ii)

From equation (i) and (ii),

$4 α β = - 20 \Rightarrow α β = - 5$

${(β^{3} - α^{3})}^{2} = {(β - α)}^{2} {(β^{2} + α^{2} + α β)}^{2}$

$= {(β - α)}^{2} {({(β + α)}^{2} - α β)}^{2}$

$= {(\sqrt{11})}^{2} {(- 9 + 5)}^{2}$

$= 11 \times 6 = 176$

Solution

Q.
Let $a, b \in C$ . Let $α, β$ be the roots of the equation $x^{2} + a x + b = 0$ . If $β - α = \sqrt{11}$ and $β^{2} - α^{2} = 3 i \sqrt{11}$ , then ${(β^{3} - α^{3})}^{2}$ is equal to : [2026]

1 160

2 176

3 194

4 187

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