Solve for x: $12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0 .$

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Solution

2abx² − (9a² − 8b²)x − 6ab = 0
⇒ 12abx² − 9a²x + 8b²x − 6ab = 0
⇒ 3ax(4bx − 3a) + 2b(4bx − 3a) = 0
⇒ (3ax + 2b)(4bx − 3a) = 0
⇒ (3ax + 2b) = 0 or (4bx − 3a) = 0
⇒ 3ax = − 2b or 4bx = 3a
⇒ $x = \frac{- 2 b}{3 a}$ or $x = \frac{3 a}{4 b}$

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