Let roots of the quadratic equation be a and b. Then, a+b=7. . . (i) a⋅b=12. . . (ii) Put b=7−a into Eq. (ii), we get a(7−a)=12 ⇒7a−a2=12 ⇒a2−7a+12=0 ⇒a2−3a−4a+12=0 ⇒a(a−3)−4(a−3)=0 ⇒(a−3)(a−4)=0 a=3 or 4 When, a=3,b=7−3=4 When, a=4,b=7−4=3 So, the smaller root is 3 and the bigger root is 4 . According to the question, New roots are
1
2
×(4) and 2×(3) i.e., 2 and 6. The resulting quadratic equation : x2−( sum of roots )x+ (product of roots )=0 ⇒x2−(2+6)x+2×6=0 ⇒x2−8x+12=0 Hence, the required quadratic equation is x2−8x+12=0