Find roots of polynomial $$ p(x) = 12x^3-54x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{ 3 \sqrt{ 2}}{ 2 }\\[1 em]x_3 &= -3 \frac{\sqrt{ 2 }}{ 2 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 6x }$ from $ 12x^3-54x $ and solve two separate equations:

$$ \begin{aligned} 12x^3-54x & = 0\\[1 em] \color{blue}{ 6x }\cdot ( 2x^2-9 ) & = 0 \\[1 em] \color{blue}{ 6x = 0} ~~ \text{or} ~~ 2x^2-9 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 2x^2-9 = 0 $ are: $ x = -3 \dfrac{\sqrt{ 2 }}{ 2 } ~ \text{and} ~ x = \dfrac{ 3 \sqrt{ 2}}{ 2 }$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

Polynomial Roots Calculator