Find roots of polynomial $$ p(x) = x^2+\frac{1}{5}x+1 $$

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

$$ \begin{aligned}x_1 &= -\frac{ 1 }{ 10 }+\frac{ 3 \sqrt{ 11}}{ 10 }i\\[1 em]x_2 &= -\frac{ 1 }{ 10 }-3 \frac{\sqrt{ 11 }}{ 10 }i \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 5 } $.

$$ \begin{aligned} x^2+\frac{1}{5}x+1 & = 0 ~~~ / \cdot \color{blue}{ 5 } \\[1 em] 5x^2+x+5 & = 0 \end{aligned} $$

Step 2:

The solutions of $ 5x^2+x+5 = 0 $ are: $ x = -\dfrac{ 1 }{ 10 }+\dfrac{ 3 \sqrt{ 11}}{ 10 }i ~ \text{and} ~ x = -\dfrac{ 1 }{ 10 }-\dfrac{ 3 \sqrt{ 11}}{ 10 }i$.

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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