$α, β$ are the roots of the equation $x^{2} + 14 x + 10 = 0.$
Find the value of ( $α^{2}$ + $β^{2}$ ).

$196$

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$206$

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$186$

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$176$

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Solution

The correct option is D
$176$

Given: $α, β$ are the roots of the equation $x^{2} + 14 x + 10 = 0.$
Using the relation between roots and coefficients,
$α + β = - 14 . . . (i)$
$α β = 10 . . . (i i)$
We know ${(α + β)}^{2} = α^{2} + β^{2} + 2 α β$
$α^{2}$ + $β^{2} = {(α + β)}^{2} - 2 α β$

$\Rightarrow α^{2}$ + $β^{2} = {(- 14)}^{2} - 2 \times 10$

$= 196 - 20$

$= 176$

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