Quadratic Equation 1329C4

1. **State the problem:** Solve the quadratic equation $$12abx^2 - \left(9a^2 - 8b^2\right)x - 6ab = 0$$ for $x$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$Ax^2 + Bx + C = 0$$, the solutions are given by $$x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$ 3. **Identify coefficients:** Here, $$A = 12ab$$ $$B = -\left(9a^2 - 8b^2\right) = -9a^2 + 8b^2$$ $$C = -6ab$$ 4. **Calculate the discriminant:** $$\Delta = B^2 - 4AC = \left(-9a^2 + 8b^2\right)^2 - 4 \times 12ab \times (-6ab)$$ 5. **Expand and simplify the discriminant:** $$\left(-9a^2 + 8b^2\right)^2 = ( -9a^2 )^2 - 2 \times 9a^2 \times 8b^2 + (8b^2)^2 = 81a^4 - 144a^2b^2 + 64b^4$$ $$-4 \times 12ab \times (-6ab) = 288 a^2 b^2$$ So, $$\Delta = 81a^4 - 144a^2b^2 + 64b^4 + 288a^2b^2 = 81a^4 + 144a^2b^2 + 64b^4$$ 6. **Recognize perfect square:** $$81a^4 + 144a^2b^2 + 64b^4 = (9a^2 + 8b^2)^2$$ 7. **Apply quadratic formula:** $$x = \frac{-B \pm \sqrt{\Delta}}{2A} = \frac{-(-9a^2 + 8b^2) \pm (9a^2 + 8b^2)}{2 \times 12ab} = \frac{9a^2 - 8b^2 \pm (9a^2 + 8b^2)}{24ab}$$ 8. **Calculate the two roots:** - For the plus sign: $$x_1 = \frac{9a^2 - 8b^2 + 9a^2 + 8b^2}{24ab} = \frac{18a^2}{24ab} = \frac{3a}{4b}$$ - For the minus sign: $$x_2 = \frac{9a^2 - 8b^2 - (9a^2 + 8b^2)}{24ab} = \frac{9a^2 - 8b^2 - 9a^2 - 8b^2}{24ab} = \frac{-16b^2}{24ab} = -\frac{2b}{3a}$$ **Final answer:** $$x = \frac{3a}{4b} \quad \text{or} \quad x = -\frac{2b}{3a}$$