Problem 4

Reduce to one fraction with the lowest possible denominator:

(3*a+2*b)/(a+b)-(25*a2-b2)/(a2-b2)=a/(2*b)

Solution

a) Write a common denominator. Since a2-b2=(a+b)*(a-b) rewrite

(3*a+2*b)/(a+b)=((3*a+2*b)*(a-b))/((a+b)*(a-b))

b) Combine the fractions:

((3*a+2*b)*(a-b)-(25*a2-b2))/((a+b)*(a-b))

c) Expand the numerator. Compute

(3*a+2*b)*(a-b)=3*a2-3*a*b+-2*b2=3*a2-a*b-2*b2

and then subtract

3*a2-a*b-2*b2-(25*a2-b2)=-22*a2-a*b-b2.

d) Thus, the expression reduces to

-(22*a2+a*b+b2)/((a+b)*(a-b))=-(22*a2+a*b+b2)/(a2-b2)

Answer: -(22*a2+a*b+b2)/(a2-b2)