Solve each of the following quadratic equations:
12abx2−(9a2−8b2)x−6ab=0
(1) Using factorization
12abx2−(9a2−8b2)x−6ab=0⇒12abx2−9xa2−8b2x−6ab=0⇒3ax(4bx−3a)+2b(4bx−3a)=0⇒(3ax+2b)(4bx−3a)=0∴3ax+2b=0 or 4bx−3a=0⇒x=−2b3a or 3a4b
(2) Using quadratic formula
12abx2−(9a2−8b2)x−6ab=0
Discriminant, D=b2−4ac=[−(9a2−8b2)]2−4(12ab)(−6ab)=81a4+144a2b2+64b4+288a2b2=81a4+144a2b2+64b4=(9a2+8b2)2≥0
As D≥0, therefore, the roots are real.
x=−b±√D2a=(9a2−8b2)±√(9a2+8b2)22(12ab)=(9a2−8b2)±(9a2+8b2)24ab=(9a2−8b2)+(9a2+8b2)24ab or (9a2−8b2)−(9a2+8b2)24ab=9a2−8b2+9a2+8b224ab or 9a2−8b2−9a2−8b224ab=18a224ab or −16b224ab=3a4b or −2b3a