Find roots of polynomial $$ p(x) = -100x-\frac{1}{20000}x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -2000000 \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 20000 } $.

$$ \begin{aligned} -100x-\frac{1}{20000}x^2 & = 0 ~~~ / \cdot \color{blue}{ 20000 } \\[1 em] -2000000x-x^2 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} -2000000x-x^2 & = 0\\[1 em] -x^2-2000000x & = 0 \end{aligned} $$

Step 3:

Factor out $ \color{blue}{ -x }$ from $ -x^2-2000000x $ and solve two separate equations:

$$ \begin{aligned} -x^2-2000000x & = 0\\[1 em] \color{blue}{ -x }\cdot ( x+2000000 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ x+2000000 & = 0 \end{aligned} $$

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

Step 4:

To find the second zero, solve equation $ x+2000000 = 0 $

$$ \begin{aligned} x+2000000 & = 0 \\[1 em] x & = -2000000 \end{aligned} $$

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