Find roots of polynomial $$ p(x) = 4t^3-5t^2-6t $$

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

$$ \begin{aligned}t_1 &= 0\\[1 em]t_2 &= 2\\[1 em]t_3 &= -\frac{ 3 }{ 4 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ t }$ from $ 4t^3-5t^2-6t $ and solve two separate equations:

$$ \begin{aligned} 4t^3-5t^2-6t & = 0\\[1 em] \color{blue}{ t }\cdot ( 4t^2-5t-6 ) & = 0 \\[1 em] \color{blue}{ t = 0} ~~ \text{or} ~~ 4t^2-5t-6 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 4t^2-5t-6 = 0 $ are: $ t = -\dfrac{ 3 }{ 4 } ~ \text{and} ~ t = 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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