If $α, β$ are roots of the quadratic equation $x^{2} - x (a - 2) - (a + 1) = 0$ when $a$ is a variable then the least value of $α^{2} + β^{2}$

5

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3

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4

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1

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Solution

The correct option is C $5$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= (a - 2)^{2} + 2 (a + 1)$
$= a^{2} - 4 a + 4 + 2 a + 2$
$= a^{2} - 2 a + 6$
$= (a - 1)^{2} + 5$
So least value of $α^{2} + β^{2}$ is 5 when $a = 1$

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