Question
$$\frac{1+x}{2}+\frac{3-x}{4} = x\cdot(9-2x)\cdot(12-2x)$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= x\cdot(9-2x)\cdot(12-2x)&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= 4x\cdot(9-2x)\cdot(12-2x)&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= 16x^3-168x^2+432x&& \text{simplify left side} \\[1 em]x+5 &= 16x^3-168x^2+432x&& \text{move all terms to the left hand side } \\[1 em]x+5-16x^3+168x^2-432x &= 0&& \text{simplify left side} \\[1 em]-16x^3+168x^2-431x+5 &= 0&& \\[1 em] \end{aligned} $$
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