$$\frac{2x+7}{3x}+\frac{5}{6} = \frac{1}{2}$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = -\dfrac{ 7 }{ 4 }-\dfrac{\sqrt{ 41 }}{ 4 } & x_3 = -\dfrac{ 7 }{ 4 }+\dfrac{\sqrt{ 41 }}{ 4 } \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{2x+7}{3x}+\frac{5}{6} &= \frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ 3x\cdot6\cdot2 }. \\[1 em]3x\cdot6\cdot2 \cdot \frac{2x+7}{3x}+3x\cdot6\cdot2\cdot\frac{5}{6} &= 3x\cdot6\cdot2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]24x^3+84x^2+30x &= 18x&& \text{move all terms to the left hand side } \\[1 em]24x^3+84x^2+30x-18x &= 0&& \text{simplify left side} \\[1 em]24x^3+84x^2+12x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 24x^{3}+84x^{2}+12x = 0 } $, first we need to factor our $ x $.
$$ 24x^{3}+84x^{2}+12x = x \left( 24x^{2}+84x+12 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The remaining roots can be found by solving equation $ 24x^{2}+84x+12 = 0$.
$ 24x^{2}+84x+12 = 0 $ is a quadratic equation.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this equation.
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