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Question 24 : The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

1. 1
2. 2
3. 3
4. 4

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Method of solving this CAT Question from Quadratic Equations

Let p and q be two roots of the equation
We have to find the minimum value of (p2 + q2)
p2 + q2 = (p + q)2 – 2pq ----(i)
Sum of the roots = $$frac{−𝑏}{𝑎}$. Therefore, p + q = -$a + 3)
Product of the roots = $$frac{c}{𝑎}$ Therefore, pq = -$a + 5)
Substituting these values back in (i)
p2 + q2 = [(-a + 3)2 – 2(-a + 5)]
p2 + q2 = a2 + 6a + 9 + 2a + 10
p2 + q2 = a2 + 8a + 19
We have to find the minimum value of a2 + 8a + 19.
By completion of squares, 19 can be split into 16 and 3
⟹ a2 + 8a + 16 + 3
⟹ (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 x2 + (a + 3)x - (a + 5) = 0 is"

Hence, the answer is 3

Choice C is the correct answer

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