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.

Answer 1

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.