Question
Find roots of polynomial $$ p(x) = \frac{1}{5}x^2-\frac{6}{5}x^4 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \dfrac{\sqrt{ 6 }}{ 6 }\\[1 em]x_3 &= - \dfrac{\sqrt{ 6 }}{ 6 } \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 5 } $.
$$ \begin{aligned} \frac{1}{5}x^2-\frac{6}{5}x^4 & = 0 ~~~ / \cdot \color{blue}{ 5 } \\[1 em] x^2-6x^4 & = 0 \end{aligned} $$Step 2:
Write polynomial in descending order
$$ \begin{aligned} x^2-6x^4 & = 0\\[1 em] -6x^4+x^2 & = 0 \end{aligned} $$Step 3:
Factor out $ \color{blue}{ -x^2 }$ from $ -6x^4+x^2 $ and solve two separate equations:
$$ \begin{aligned} -6x^4+x^2 & = 0\\[1 em] \color{blue}{ -x^2 }\cdot ( 6x^2-1 ) & = 0 \\[1 em] \color{blue}{ -x^2 = 0} ~~ \text{or} ~~ 6x^2-1 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 4:
The solutions of $ 6x^2-1 = 0 $ are: $ x = - \dfrac{\sqrt{ 6 }}{ 6 } ~ \text{and} ~ x = \dfrac{\sqrt{ 6 }}{ 6 }$.
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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