Find roots of polynomial $$ p(x) = x^4-13x^6 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{\sqrt{ 13 }}{ 13 }\\[1 em]x_3 &= - \frac{\sqrt{ 13 }}{ 13 } \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} x^4-13x^6 & = 0\\[1 em] -13x^6+x^4 & = 0 \end{aligned} $$

Step 2:

Factor out $ \color{blue}{ -x^4 }$ from $ -13x^6+x^4 $ and solve two separate equations:

$$ \begin{aligned} -13x^6+x^4 & = 0\\[1 em] \color{blue}{ -x^4 }\cdot ( 13x^2-1 ) & = 0 \\[1 em] \color{blue}{ -x^4 = 0} ~~ \text{or} ~~ 13x^2-1 & = 0 \end{aligned} $$

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

Step 3:

The solutions of $ 13x^2-1 = 0 $ are: $ x = - \dfrac{\sqrt{ 13 }}{ 13 } ~ \text{and} ~ x = \dfrac{\sqrt{ 13 }}{ 13 }$.

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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