First slide

Theory of equations

Question

$If α, β are the roots of the equation x^{2} - (1 + n^{2}) x + \frac{1}{2} (1 + n^{2} + n^{4}) = 0 then the value of α^{2} + β^{2} is$

Moderate

A

$2 n$
B

$n^{3}$
C

$n^{2}$
D

$2 n^{2}$

Solution

$α + β = 1 + n^{2}; αβ = \frac{1}{2} (1 + n^{2} + n^{4})$
$∴ α^{2} + β^{2} = {(α + β)}^{2} - 2 αβ$
$= {(1 + n^{2})}^{2} - 2 . \frac{1}{2} (1 + n^{2} + n^{4})$
$= 1 + n^{4} + 2 n^{2} - 1 - n^{2} - n^{4} \Rightarrow α^{2} + β^{2} = n^{2} .$

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