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Question 24 : The minimum possible value of the sum of the squares of the roots of the equation x^{2} + (a + 3)x - (a + 5) = 0 is

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Let p and q be two roots of the equation

We have to find the minimum value of (p^{2} + q^{2})

p^{2} + q^{2} = (p + q)^{2} โ 2pq ----(i)

Sum of the roots = \\frac{โ๐}{๐}).

Therefore, p + q = -(a + 3)

Product of the roots = \\frac{c}{๐})

Therefore, pq = -(a + 5)

Substituting these values back in (i)

p^{2} + q^{2} = [(-a + 3)^{2} โ 2(-a + 5)]

p^{2} + q^{2} = a^{2} + 6a + 9 + 2a + 10

p^{2} + q^{2} = a^{2} + 8a + 19

We have to find the minimum value of a^{2} + 8a + 19.

By completion of squares, 19 can be split into 16 and 3

โน a^{2} + 8a + 16 + 3

โน (a + 4)^{2} + 3

The minimum possible value is 3 when a = -4.

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

Choice C is the correct answer

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