Question
Find roots of polynomial $$ p(x) = \frac{4}{5}x^3-\frac{17}{10}x^2+\frac{6}{5}x $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \dfrac{ 17 }{ 16 }+\dfrac{\sqrt{ 95 }}{ 16 }i\\[1 em]x_3 &= \dfrac{ 17 }{ 16 }- \dfrac{\sqrt{ 95 }}{ 16 }i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 10 } $.
$$ \begin{aligned} \frac{4}{5}x^3-\frac{17}{10}x^2+\frac{6}{5}x & = 0 ~~~ / \cdot \color{blue}{ 10 } \\[1 em] 8x^3-17x^2+12x & = 0 \end{aligned} $$Step 2:
Factor out $ \color{blue}{ x }$ from $ 8x^3-17x^2+12x $ and solve two separate equations:
$$ \begin{aligned} 8x^3-17x^2+12x & = 0\\[1 em] \color{blue}{ x }\cdot ( 8x^2-17x+12 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 8x^2-17x+12 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 3:
The solutions of $ 8x^2-17x+12 = 0 $ are: $ x = \dfrac{ 17 }{ 16 }+\dfrac{\sqrt{ 95 }}{ 16 }i ~ \text{and} ~ x = \dfrac{ 17 }{ 16 }-\dfrac{\sqrt{ 95 }}{ 16 }i$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.
This page was created using
Polynomial Roots Calculator