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

2

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43

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5

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7

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Solution

The correct option is C $5$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= (a - 2)^{2} + 2 (a + 1)$
$= a^{2} - 4 a + 4 + 2 a + 2$
$= a^{2} - 2 a + 6$
$= (a - 1)^{2} + 5$
Minimum value is at $a = 1$ .
Value of expression at $a = 1$ is $5$ .

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