The minimum possible value of the sum of squares of the roots of the equation $x^{2} + (a + 3) x - (a + 5) = 0$ is

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Solution

The correct option is C
3

Given equation is $x^{2} + (a + 3) x - (a + 5) = 0$
Let ‘p’ and ‘q’ be the roots of the given equation.
Then, p + q = -(a+3) and pq = -(a+5)
So, $p^{2} + q^{2} = (p + q)^{2} - 2 p q$
$\Rightarrow [- (a + 3)]^{2} - 2 [- (a + 5)]$
$\Rightarrow (a + 3)^{2} + 2 (a + 5)$
$\Rightarrow a^{2} + 9 + 6 a + 2 a + 10$
$\Rightarrow a^{2} + 8 a + 19$
$\Rightarrow (a + 4)^{2} + 3$
Minimum possible value is 3, as $(a + 4)^{2}$ is never negative.

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