Question
Solve the following quadratic equation:
$ 12 a b x^2-\left(9 a^2-8 b^2\right) x-6 a b=0$
$ 12 a b x^2-\left(9 a^2-8 b^2\right) x-6 a b=0$
✓
Answer
$ 12 a b x^2-\left(9 a^2-8 b^2\right) x-6 a b=0$
$\Rightarrow 12 a b x^2-9 a^2 x+8 b^2 x-6 a b=0$
$\Rightarrow 3 a x(4 b x-3 a)+2 b(4 b x-3 a)=0$
$\Rightarrow(4 b x-3 a)(3 a x+2 b)=0$
$\Rightarrow(4 b x-3 a)=0 \text { or }(3 a x+2 b)=0$
$\Rightarrow 4 b x=3 a \text { or } 3 a x=-2 b$
$\Rightarrow\text{x}=\frac{\text{3a}}{\text{4b}}$ or $\text{x}=\frac{-\text{2b}^2}{\text{3a}}$
Hence, $\frac{\text{3a}}{\text{4b}}$ and $\frac{-\text{2b}}{\text{3a}}$ are the roots of the given equation.
$\Rightarrow 12 a b x^2-9 a^2 x+8 b^2 x-6 a b=0$
$\Rightarrow 3 a x(4 b x-3 a)+2 b(4 b x-3 a)=0$
$\Rightarrow(4 b x-3 a)(3 a x+2 b)=0$
$\Rightarrow(4 b x-3 a)=0 \text { or }(3 a x+2 b)=0$
$\Rightarrow 4 b x=3 a \text { or } 3 a x=-2 b$
$\Rightarrow\text{x}=\frac{\text{3a}}{\text{4b}}$ or $\text{x}=\frac{-\text{2b}^2}{\text{3a}}$
Hence, $\frac{\text{3a}}{\text{4b}}$ and $\frac{-\text{2b}}{\text{3a}}$ are the roots of the given equation.
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