Find the value of `a` for which the sum of the squares of the roots of the equation `x^2-(a-2)x-a-1=0` assumes the least value.

Answer

a = 1

Explanation

Let `alpha , beta` be the roots of the given equation . Then,
`alpha + beta ` = a - 2 and `alpha beta = - (a + 1)`
Now, `alpha^(2) + beta^(2) =( alpha + beta)^(2) - 2ab`
`= (a - 2)^(2) + 2 (a + 1)`
`a^(2) - 2a + 6`
`(a - 1)^(2) + 5`
Clearly `alpha^(2) + beta^(2) ge 5 .` So, the minimum value of `alpha^(2) + beta^(2)` is 5 which it attains at a = 1