Find the roots of the given equation, if they exist, by applying the quadratic formula:
$3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0$

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Solution

$3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0$

$a = 3 a^{2} b = 8 a b c = 4 b^{2}$

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 8 a b \pm \sqrt{(8 a b)^{2} - 4 \times 3 a^{2} \times 4 b^{2}}}{2 \times 3 a^{2}} = \frac{- 8 a b \pm \sqrt{64 a^{2} b^{2} - 48 a^{2} b^{2}}}{6 a^{2}} = \frac{- 8 a b \pm \sqrt{16 a^{2} b^{2}}}{6 a^{2}} = \frac{- 8 a b \pm 4 a b}{6 a^{2}} = \frac{- 8 a b + 4 a b}{6 a^{2}} o r \frac{- 8 a b - 4 a b}{6 a^{2}} = \frac{- 4 a b}{6 a^{2}} o r \frac{- 12 a b}{6 a^{2}} x = \frac{- 2 b}{3 a} o r \frac{- 2 b}{a}$

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$3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0$