$$\frac{x^2+3}{7}x = \frac{x+1}{6}$$
Answer
$$ \begin{matrix}x_1 = 0.54706 & x_2 = -0.27353+1.4345i & x_3 = -0.27353-1.4345i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x^2+3}{7}x &= \frac{x+1}{6}&& \text{multiply ALL terms by } \color{blue}{ 42 }. \\[1 em]42 \cdot \frac{x^2+3}{7}x &= 42 \cdot \frac{x+1}{6}&& \text{cancel out the denominators} \\[1 em]6x^3+18x &= 7x+7&& \text{move all terms to the left hand side } \\[1 em]6x^3+18x-7x-7 &= 0&& \text{simplify left side} \\[1 em]6x^3+11x-7 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver