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Question
Find the value of a such that the sum of the squares of the roots of the equation x2 – (a – 2)x – (a + 1) = 0 is least.
Solution
Let α, β be the roots of the equation.
Therefore, α + β = a – 2 and αβ = –( a + 1)
Now α2 + β2 = (α + β)2 – 2αβ
= (a – 2)2 + 2(a + 1)
= (a – 1)2 + 5
Therefore, α2 + β2 will be minimum if (a – 1)2 = 0, i.e., a = 1.
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