X^4+4x-3=0

There's no nice way to solve this as far as I can see.  But we can use the solver here to do it:

 

$${{\mathtt{x}}}^{{\mathtt{4}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{3}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\frac{{\mathtt{8}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}}{\mathtt{\,-\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}\\
{\mathtt{x}} = {\frac{{\sqrt{{\frac{{\mathtt{8}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}}{\mathtt{\,-\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}\\
{\mathtt{x}} = {\frac{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{8}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}}{\mathtt{\,-\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}}}}}}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{8}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}}{\mathtt{\,-\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}}}}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\sqrt{{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{4}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{1.784\: \!357\: \!981\: \!032\: \!616\: \!8}}\\
{\mathtt{x}} = {\mathtt{0.692\: \!504\: \!842\: \!571\: \!842\: \!6}}\\
{\mathtt{x}} = {\mathtt{0.545\: \!926\: \!569\: \!230\: \!387\: \!1}}{\mathtt{\,-\,}}{\mathtt{1.459\: \!377\: \!949\: \!580\: \!500\: \!1}}{i}\\
{\mathtt{x}} = {\mathtt{0.545\: \!926\: \!569\: \!230\: \!387\: \!1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.459\: \!377\: \!949\: \!580\: \!500\: \!1}}{i}\\
\end{array} \right\}$$