Find roots of polynomial $$ p(x) = 12x^3+18x^2-2x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.1039\\[1 em]x_3 &= -1.6039 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 2x }$ from $ 12x^3+18x^2-2x $ and solve two separate equations:

$$ \begin{aligned} 12x^3+18x^2-2x & = 0\\[1 em] \color{blue}{ 2x }\cdot ( 6x^2+9x-1 ) & = 0 \\[1 em] \color{blue}{ 2x = 0} ~~ \text{or} ~~ 6x^2+9x-1 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 6x^2+9x-1 = 0 $ are: $ x = -\dfrac{ 3 }{ 4 }-\dfrac{\sqrt{ 105 }}{ 12 } ~ \text{and} ~ x = -\dfrac{ 3 }{ 4 }+\dfrac{\sqrt{ 105 }}{ 12 }$.

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

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