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

The equation is x2−(a−2)x−a−1=0 Let the roots are α,β Sum of the roots : α+β=a−2,   Product of the roots : αβ=−(a+1) The least value of the sum of the squares of the roots of the equation isα2+β2=(α+β)2−2αβ =(a−2)2+2a+2 =a2−2a+6=(a−1)2+5≥5 Hence least (α2+β2)=5 which is obtained at a=1 .