$$\frac{1+x}{2}+\frac{3-x}{4} = x^4-5x^3-x^2+25x-20$$
Answer
$$ \begin{matrix}x_1 = 1.14604 & x_2 = -2.24366 & x_3 = 2.03342 \\[1 em] x_4 = 4.06421 & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= x^4-5x^3-x^2+25x-20&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= 4x^4-4\cdot5x^3-4x^2+4\cdot25x-4\cdot20&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= 4x^4-20x^3-4x^2+100x-80&& \text{simplify left side} \\[1 em]x+5 &= 4x^4-20x^3-4x^2+100x-80&& \text{move all terms to the left hand side } \\[1 em]x+5-4x^4+20x^3+4x^2-100x+80 &= 0&& \text{simplify left side} \\[1 em]-4x^4+20x^3+4x^2-99x+85 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
This page was created using
Polynomial Equations Solver