Find roots of polynomial $$ p(x) = 6n^2+4n $$

The roots of polynomial $ p(n) $ are:

$$ \begin{aligned}n_1 &= 0\\[1 em]n_2 &= -\frac{ 2 }{ 3 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 2n }$ from $ 6n^2+4n $ and solve two separate equations:

$$ \begin{aligned} 6n^2+4n & = 0\\[1 em] \color{blue}{ 2n }\cdot ( 3n+2 ) & = 0 \\[1 em] \color{blue}{ 2n = 0} ~~ \text{or} ~~ 3n+2 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ n = 0 } $. Use second equation to find the remaining roots.

Step 2:

To find the second zero, solve equation $ 3n+2 = 0 $

$$ \begin{aligned} 3n+2 & = 0 \\[1 em] 3 \cdot n & = -2 \\[1 em] n & = - \frac{ 2 }{ 3 } \end{aligned} $$

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