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Your input: solve the quadratic equation $$$2 n^{2} + 3 n + 1 = 0$$$ by using quadratic formula.

The standard quadratic equation has the form $$$an^2+bn+c=0$$$.

In our case, $$$a=2$$$, $$$b=3$$$, $$$c=1$$$.

Now, find the discriminant using the formula $$$D=b^2-4ac$$$: $$$D=3^2-4\cdot 2 \cdot 1=1$$$.

Find the roots of the equation using the formulas $$$n_1=\frac{-b-\sqrt{D}}{2a}$$$ and $$$n_2=\frac{-b+\sqrt{D}}{2a}$$$

$$$n_1=\frac{-3-\sqrt{1}}{2\cdot 2}=-1$$$ and $$$n_2=\frac{-3+\sqrt{1}}{2\cdot 2}=- \frac{1}{2}$$$

Answer: $$$n_1=-1$$$; $$$n_2=- \frac{1}{2}$$$