Find roots of polynomial $$ p(x) = -18x^4-33x^3-6x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.2047\\[1 em]x_3 &= -1.6287 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -3x^2 }$ from $ -18x^4-33x^3-6x^2 $ and solve two separate equations:

$$ \begin{aligned} -18x^4-33x^3-6x^2 & = 0\\[1 em] \color{blue}{ -3x^2 }\cdot ( 6x^2+11x+2 ) & = 0 \\[1 em] \color{blue}{ -3x^2 = 0} ~~ \text{or} ~~ 6x^2+11x+2 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 6x^2+11x+2 = 0 $ are: $ x = -\dfrac{ 11 }{ 12 }-\dfrac{\sqrt{ 73 }}{ 12 } ~ \text{and} ~ x = -\dfrac{ 11 }{ 12 }+\dfrac{\sqrt{ 73 }}{ 12 }$.

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

Polynomial Roots Calculator