Find roots of polynomial $$ p(x) = 3x^2-x-\frac{6}{4}x^2-9 $$

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

$$ \begin{aligned}x_1 &= 2.8054\\[1 em]x_2 &= -2.1387 \end{aligned} $$

Step 1:

Get rid of fractions by multipling equation by $ \color{blue}{ 4 } $.

$$ \begin{aligned} 3x^2-x-\frac{6}{4}x^2-9 & = 0 ~~~ / \cdot \color{blue}{ 4 } \\[1 em] 12x^2-4x-6x^2-36 & = 0 \end{aligned} $$

Step 2:

Combine like terms:

$$ \color{blue}{12x^2} -4x \color{blue}{-6x^2} -36 = \color{blue}{6x^2} -4x-36 $$

Step 3:

The solutions of $ 6x^2-4x-36 = 0 $ are: $ x = \dfrac{ 1 }{ 3 }-\dfrac{\sqrt{ 55 }}{ 3 } ~ \text{and} ~ x = \dfrac{ 1 }{ 3 }+\dfrac{\sqrt{ 55 }}{ 3 }$.

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