Find roots of polynomial $$ p(x) = h^3+18h^2-45h $$

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

$$ \begin{aligned}h_1 &= 0\\[1 em]h_2 &= 2.225\\[1 em]h_3 &= -20.225 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ h }$ from $ h^3+18h^2-45h $ and solve two separate equations:

$$ \begin{aligned} h^3+18h^2-45h & = 0\\[1 em] \color{blue}{ h }\cdot ( h^2+18h-45 ) & = 0 \\[1 em] \color{blue}{ h = 0} ~~ \text{or} ~~ h^2+18h-45 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ h^2+18h-45 = 0 $ are: $ h = -9-3 \sqrt{ 14 } ~ \text{and} ~ h = -9+3 \sqrt{ 14 }$.

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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