First slide

Theory of equations

Question

Two complex numbers $a$ and $β$ are such
$that α + β = 2 and α^{4} + β^{4} = 272, then the quadratic equation$
$whose roots are α and β can be$

Moderate

A

$x^{2} - 2 x - 16 = 0$
B

$x^{2} - 2 x + 12 = 0$
C

$x^{2} - 2 x - 8 = 0$
D

none of these

Solution

$α + β = 2 \Rightarrow α^{2} + β^{2} + 2 α β = 4$
$\begin{array}{l} \Rightarrow α^{2} + β^{2} = 4 - 2 α β \Rightarrow {(α^{2} + β^{2})}^{2} = 16 - 16 α β + 4 α^{2} β^{2} \\ \Rightarrow α^{4} + β^{4} + 2 α^{2} β^{2} = 16 - 16 α β + 4 α^{2} β^{2} \\ \Rightarrow 272 + 2 (α β)^{2} = 16 - 16 α β + 4 (α β)^{2} \\ \Rightarrow 2 (α β)^{2} - 16 (α β) - 256 = 0 \\ \Rightarrow (α β)^{2} - 8 (α β) - 128 = 0 \\ \Rightarrow (α β - 16) (α β + 8) = 0 \\ \Rightarrow α β = 16 or α β = - 8 \end{array}$
Thus, required equation is either
or $\begin{array}{l} x^{2} - 2 x + 16 = 0 \\ x^{2} - 2 x - 8 = 0 \end{array}$
$∴$ Required answer is (c).

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