Given quadratic equation, x2−(a−2)x−a−1=0 Let α and β are the roots of the equation. ∴α+β=a−2 and αβ=−a−1 Now α2+β2=(α+β)2−2αβ ‌⇒α2+β2=(a−2)2+2(a+1) ‌⇒α2+β2=a2−2a+6 ‌⇒α2+β2=(a−1)2+5 ⇒ The value of α2+β2 will be minimum, when a−1=0 ⇒a=1