The solutions of x for which the following quadratic equation holds 12abx2-9a2-8b2x-6ab=0 are

# The solutions of x for which the following quadratic equation holds $12\mathit{ab}{x}^{2}-\left(9{a}^{2}-8{b}^{2}\right)x-6\mathit{ab}=0$ are

1. A
2. B
3. C
4. D
None of these

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### Solution:

Given equation: $12\mathit{ab}{x}^{2}-\left(9{a}^{2}-8{b}^{2}\right)x-6\mathit{ab}=0$ On comparing the given equation with $a{x}^{2}+\mathit{bx}+c=0,$ we get
a = $12\mathit{ab}$, b = $-\left(9{a}^{2}-8{b}^{2}\right)$ and c = $-6\mathit{ab}$
$x=\frac{-b±\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$
$x=\frac{-\left[-\left(9{a}^{2}-8{b}^{2}\right)\right]±\sqrt{{\left[-\left(9{a}^{2}-8{b}^{2}\right)\right]}^{2}-4\left(12\mathit{ab}\right)\left(-6\mathit{ab}\right)}}{2\left(12\mathit{ab}\right)}$
$x=\frac{\left(9{a}^{2}-8{b}^{2}\right)±\sqrt{81{a}^{2}+64{b}^{2}-144{a}^{2}{b}^{2}+288{a}^{2}{b}^{2}}}{2\left(12\mathit{ab}\right)}$
$x=\frac{\left(9{a}^{2}-8{b}^{2}\right)±\sqrt{81{a}^{2}+64{b}^{2}+144{a}^{2}{b}^{2}}}{2\left(12\mathit{ab}\right)}$
$x=\frac{\left(9{a}^{2}-8{b}^{2}\right)±\sqrt{{\left[\left(9{a}^{2}+8{b}^{2}\right)\right]}^{2}}}{2\left(12\mathit{ab}\right)}$
$x=\frac{\left(9{a}^{2}-8{b}^{2}\right)±\left(9{a}^{2}+8{b}^{2}\right)}{2\left(12\mathit{ab}\right)}$
$x=\frac{\left(9{a}^{2}-8{b}^{2}\right)±\left(9{a}^{2}+8{b}^{2}\right)}{2\left(12\mathit{ab}\right)}$
and
and $x=\frac{-16{b}^{2}}{24\mathit{ab}}$
and $x=\frac{-2b}{3a}$
Therefore, 2 is the correct option.

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