Find roots of polynomial $$ p(x) = 27x^2-12x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{ 4 }{ 9 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 3x }$ from $ 27x^2-12x $ and solve two separate equations:

$$ \begin{aligned} 27x^2-12x & = 0\\[1 em] \color{blue}{ 3x }\cdot ( 9x-4 ) & = 0 \\[1 em] \color{blue}{ 3x = 0} ~~ \text{or} ~~ 9x-4 & = 0 \end{aligned} $$

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

Step 2:

To find the second zero, solve equation $ 9x-4 = 0 $

$$ \begin{aligned} 9x-4 & = 0 \\[1 em] 9 \cdot x & = 4 \\[1 em] x & = \frac{ 4 }{ 9 } \end{aligned} $$

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