Find roots of polynomial $$ p(x) = 2s^3+7s^2-8s $$

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

$$ \begin{aligned}s_1 &= 0\\[1 em]s_2 &= 0.9075\\[1 em]s_3 &= -4.4075 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ s }$ from $ 2s^3+7s^2-8s $ and solve two separate equations:

$$ \begin{aligned} 2s^3+7s^2-8s & = 0\\[1 em] \color{blue}{ s }\cdot ( 2s^2+7s-8 ) & = 0 \\[1 em] \color{blue}{ s = 0} ~~ \text{or} ~~ 2s^2+7s-8 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 2s^2+7s-8 = 0 $ are: $ s = -\dfrac{ 7 }{ 4 }-\dfrac{\sqrt{ 113 }}{ 4 } ~ \text{and} ~ s = -\dfrac{ 7 }{ 4 }+\dfrac{\sqrt{ 113 }}{ 4 }$.

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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