Find roots of polynomial $$ p(x) = x+\frac{3}{2}x^2-6 $$

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

$$ \begin{aligned}x_1 &= 1.6943\\[1 em]x_2 &= -2.3609 \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.

$$ \begin{aligned} x+\frac{3}{2}x^2-6 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2x+3x^2-12 & = 0 \end{aligned} $$

Step 2:

Write polynomial in descending order

$$ \begin{aligned} 2x+3x^2-12 & = 0\\[1 em] 3x^2+2x-12 & = 0 \end{aligned} $$

Step 3:

The solutions of $ 3x^2+2x-12 = 0 $ are: $ x = -\dfrac{ 1 }{ 3 }-\dfrac{\sqrt{ 37 }}{ 3 } ~ \text{and} ~ x = -\dfrac{ 1 }{ 3 }+\dfrac{\sqrt{ 37 }}{ 3 }$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

Polynomial Roots Calculator