Given: |x2−x−6|=x+2 We can factorize x2−x−6 as (x−3)(x+2) ⇒|(x−3)(x+2)|={
(x−3)(x+2)
if (x−3)(x+2)≥0
−(x−3)(x+2)
if (x−3)(x+2)<0.
Case −1: If (x−3)(x+2)≥0 ⇒x∈(−∞,−2]∪[3,∞) ⇒x2−x−6=x+2 ⇒x2−2x−8=0 ⇒(x−4)(x+2)=0 ⇒x=4 or −2∈(−∞,−2]∪[3,∞) So, the roots of the given quadratic equation are 4 and −2. Case −2 : If (x−3)(x+2)<0 ⇒x∈[−2,3] ⇒−(x2−x−6)=x+2 ⇒x2−4=0 ⇒x=2 or −2∈[−2,3] So, the roots of the given quadratic equation are 2 and −2 Hence, the roots of the given quadratic equation are −2,2 and 4.