Find roots of polynomial $$ p(x) = 12x^3+9x-30x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{\sqrt{ 7 }}{ 2 }\\[1 em]x_3 &= - \frac{\sqrt{ 7 }}{ 2 } \end{aligned} $$

Step 1:

Combine like terms:

$$ 12x^3+ \color{blue}{9x} \color{blue}{-30x} = 12x^3 \color{blue}{-21x} $$

Step 2:

Factor out $ \color{blue}{ 3x }$ from $ 12x^3-21x $ and solve two separate equations:

$$ \begin{aligned} 12x^3-21x & = 0\\[1 em] \color{blue}{ 3x }\cdot ( 4x^2-7 ) & = 0 \\[1 em] \color{blue}{ 3x = 0} ~~ \text{or} ~~ 4x^2-7 & = 0 \end{aligned} $$

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

Step 3:

The solutions of $ 4x^2-7 = 0 $ are: $ x = - \dfrac{\sqrt{ 7 }}{ 2 } ~ \text{and} ~ x = \dfrac{\sqrt{ 7 }}{ 2 }$.

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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