+0

# [ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336

0
1270
3

[ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336

Aug 9, 2014

#1
+117762
+10

[ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336

$$[ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{(3+x)}\right]\div \frac{(1-x)}{(3+x)} = 0.2336\\\\ \mbox{It should be noted straight off that }3+x\ne0\;so\;x\ne-3\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{(3+x)}\right]\times \frac{(3+x)}{(1-x)} = 0.2336\qquad \mbox{the (x+3) cancels,and }x\ne1\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{1}\right]\times \frac{1}{(1-x)} = 0.2336\qquad \mbox{}\\\\ \frac{x(2+x)}{(3+x)(1-x)} = 0.2336\qquad \mbox{}\\\\ \frac{x(2+x)}{(3+x)(1-x)} = 0.2336\qquad \\\\ \mbox{Now multiply everything by the lowest common denominator (3+x)(1-x) to get rid of the fraction}\\\\ x(2+x) = 0.2336(3+x)(1-x)\qquad \\\\$$

you can keep going from here and solve it with the quadratic formula or you can just plug your initial equation into the site calc and get it to do it all for you.

(x/(3+x) * (2+x)/(3+x) ) / (1-x)/(3+x) = 0.2336

change the square brackets to round ones and change your decimal comma to a decimal point and just plug it in.

$${\frac{{\frac{\left({\frac{{\frac{{\mathtt{x}}}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}}{\mathtt{\,\times\,}}\left({\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}}\right)}{\left({\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{x}}\right)}}}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}} = {\mathtt{0.233\: \!6}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\frac{{\sqrt{{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{3.318\: \!755\: \!885\: \!933\: \!879\: \!7}}{\mathtt{\,-\,}}{\mathtt{2.004\: \!784\: \!814\: \!651\: \!891\: \!5}}{i}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{3.318\: \!755\: \!885\: \!933\: \!879\: \!7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.004\: \!784\: \!814\: \!651\: \!891\: \!5}}{i}\\ {\mathtt{x}} = -{\mathtt{2.184\: \!609\: \!027\: \!146\: \!14}}\\ {\mathtt{x}} = {\mathtt{0.822\: \!120\: \!799\: \!013\: \!899\: \!5}}\\ \end{array} \right\}$$

My goodness that looks horrible!

Let's see what the graph looks like and get the answer/s from there.

Alright there are your 2 answers.  I might keep going with the quadratic formula and get them that way.

$$x(2+x) = 0.2336(3+x)(1-x)\qquad \\\\ x^2+2x = 0.2336(3-3x+x-x^2)\qquad \\\\ x^2+2x = 0.2336(-x^2-2x+3)\qquad \\\\$$

I could just finish this by hand but I'mm goint to let the web 2 calc do it for me.

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{0.233\: \!6}}{\mathtt{\,\times\,}}\left({\mathtt{\,-\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}\right) \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{103\,571}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{257}}\right)}{{\mathtt{257}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{103\,571}}}}{\mathtt{\,-\,}}{\mathtt{257}}\right)}{{\mathtt{257}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{2.252\: \!235\: \!355\: \!360\: \!168\: \!6}}\\ {\mathtt{x}} = {\mathtt{0.252\: \!235\: \!355\: \!360\: \!168\: \!6}}\\ \end{array} \right\}$$

There you go, that is better!

These 2 answers would have been included in the original calculator ouput but some other imaginary roots were included as well.  You probably do not need to worry about those.

Aug 10, 2014

#1
+117762
+10

[ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336

$$[ x/(3+x) * (2+x)/(3+x) ] / (1-x)/(3+x) = 0,2336\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{(3+x)}\right]\div \frac{(1-x)}{(3+x)} = 0.2336\\\\ \mbox{It should be noted straight off that }3+x\ne0\;so\;x\ne-3\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{(3+x)}\right]\times \frac{(3+x)}{(1-x)} = 0.2336\qquad \mbox{the (x+3) cancels,and }x\ne1\\\\ \left[\frac{x}{(3+x)} \times \frac{(2+x)}{1}\right]\times \frac{1}{(1-x)} = 0.2336\qquad \mbox{}\\\\ \frac{x(2+x)}{(3+x)(1-x)} = 0.2336\qquad \mbox{}\\\\ \frac{x(2+x)}{(3+x)(1-x)} = 0.2336\qquad \\\\ \mbox{Now multiply everything by the lowest common denominator (3+x)(1-x) to get rid of the fraction}\\\\ x(2+x) = 0.2336(3+x)(1-x)\qquad \\\\$$

you can keep going from here and solve it with the quadratic formula or you can just plug your initial equation into the site calc and get it to do it all for you.

(x/(3+x) * (2+x)/(3+x) ) / (1-x)/(3+x) = 0.2336

change the square brackets to round ones and change your decimal comma to a decimal point and just plug it in.

$${\frac{{\frac{\left({\frac{{\frac{{\mathtt{x}}}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}}{\mathtt{\,\times\,}}\left({\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}}\right)}{\left({\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{x}}\right)}}}{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)}} = {\mathtt{0.233\: \!6}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\frac{{\sqrt{{\mathtt{\,-\,}}{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ {\mathtt{x}} = {\frac{{\sqrt{{\frac{{\mathtt{14\,508}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}}{\mathtt{\,-\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{704\,375}}}{\left({\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{502}}}{{\mathtt{219}}}}}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\sqrt{{\mathtt{191\,844}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{219\,876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\mathtt{704\,375}}}}}{\left({\mathtt{876}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{625}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{6\,484\,410\,131}}}}}{\left({\mathtt{219}}{\mathtt{\,\times\,}}{{\mathtt{146}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{10\,930\,245\,625}}}{{\mathtt{84\,027\,672}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}\right)}}{\mathtt{\,-\,}}{\mathtt{2}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{3.318\: \!755\: \!885\: \!933\: \!879\: \!7}}{\mathtt{\,-\,}}{\mathtt{2.004\: \!784\: \!814\: \!651\: \!891\: \!5}}{i}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{3.318\: \!755\: \!885\: \!933\: \!879\: \!7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.004\: \!784\: \!814\: \!651\: \!891\: \!5}}{i}\\ {\mathtt{x}} = -{\mathtt{2.184\: \!609\: \!027\: \!146\: \!14}}\\ {\mathtt{x}} = {\mathtt{0.822\: \!120\: \!799\: \!013\: \!899\: \!5}}\\ \end{array} \right\}$$

My goodness that looks horrible!

Let's see what the graph looks like and get the answer/s from there.

Alright there are your 2 answers.  I might keep going with the quadratic formula and get them that way.

$$x(2+x) = 0.2336(3+x)(1-x)\qquad \\\\ x^2+2x = 0.2336(3-3x+x-x^2)\qquad \\\\ x^2+2x = 0.2336(-x^2-2x+3)\qquad \\\\$$

I could just finish this by hand but I'mm goint to let the web 2 calc do it for me.

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{0.233\: \!6}}{\mathtt{\,\times\,}}\left({\mathtt{\,-\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}\right) \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{103\,571}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{257}}\right)}{{\mathtt{257}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{103\,571}}}}{\mathtt{\,-\,}}{\mathtt{257}}\right)}{{\mathtt{257}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{2.252\: \!235\: \!355\: \!360\: \!168\: \!6}}\\ {\mathtt{x}} = {\mathtt{0.252\: \!235\: \!355\: \!360\: \!168\: \!6}}\\ \end{array} \right\}$$

There you go, that is better!

These 2 answers would have been included in the original calculator ouput but some other imaginary roots were included as well.  You probably do not need to worry about those.

Melody Aug 10, 2014
#2
+124524
+5

WolframAlpha gets a slightly different result from Melody's answer...I suspect the reason is in the way that WA is interpreting the multiple divisions of the two functions outside the brackets....

(The only change I made to the original problem was to subtract .2336 from both sides....)

Whatever.....I think the point here is that this one would be a tough one to evaluate by hand.......Based on the original form of the problem, it would be difficult to intuit how many "real' and/or "non-real" solutions there might be......

Aug 10, 2014
#3
0

thank you both !

Aug 10, 2014