Find roots of polynomial $$ p(x) = 3t+\frac{49}{10}t^2 $$

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

$$ \begin{aligned}t_1 &= 0\\[1 em]t_2 &= -\frac{ 30 }{ 49 } \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 10 } $.

$$ \begin{aligned} 3t+\frac{49}{10}t^2 & = 0 ~~~ / \cdot \color{blue}{ 10 } \\[1 em] 30t+49t^2 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} 30t+49t^2 & = 0\\[1 em] 49t^2+30t & = 0 \end{aligned} $$

Step 3:

Factor out $ \color{blue}{ t }$ from $ 49t^2+30t $ and solve two separate equations:

$$ \begin{aligned} 49t^2+30t & = 0\\[1 em] \color{blue}{ t }\cdot ( 49t+30 ) & = 0 \\[1 em] \color{blue}{ t = 0} ~~ \text{or} ~~ 49t+30 & = 0 \end{aligned} $$

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

Step 4:

To find the second zero, solve equation $ 49t+30 = 0 $

$$ \begin{aligned} 49t+30 & = 0 \\[1 em] 49 \cdot t & = -30 \\[1 em] t & = - \frac{ 30 }{ 49 } \end{aligned} $$

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