The value of  a for which the sum of the squares of the roots of the equation x2−(a−2)x−a−1=0 assumes the least value, is

Let α,β  be the roots of the given equation. Then,a+β=a−2 and aβ=−(a+1)∴ α2+β2=(α+β)2−2αβ=(a−2)2+2(a+1)⇒ α2+β2=a2−2a+6=(a−1)2+5Clearly,  α2+β2≥5 So, the minimum value of α2+β2 is which it attains at a=1.