Answer:
[tex]x=\frac{3}{2} \pm i\sqrt{\frac{11}{4}-\frac{e^{2} }{3} }[/tex]
Step-by-step explanation:
First, cancel logarithms by taking exp of both sides:
[tex]e^{3*(x^{2}-3x+5) }=e^{2} \\ 3*(x^{2}-3x+5)=e^{2}[/tex]
Divide both sides by 3 and then substract 5 from both sides:
[tex]x^{2} -3x=\frac{e^{2} }{3} -5[/tex]
Add 9/4 to both sides in order to write the left side as a square:
[tex]x^{2} -3x+\frac{9}{4} =\frac{e^{2} }{3} -\frac{11}{4} \\(x-\frac{3}{2}) ^{2} =\frac{e^{2} }{3} -\frac{11}{4}[/tex]
Express the right side as:
[tex](-1)*(\frac{11}{4}-\frac{e^{2} }{3} )[/tex]
Now take square root of both sides, keep in mind that: [tex]i=\sqrt{-1}[/tex]
[tex]x-\frac{3}{2}=\pm \sqrt{-1} *\sqrt{\frac{11}{4}-\frac{e^{2} }{3}[/tex]
Finally, add 3/2 to both sides:
[tex]x=\frac{3}{2} \pm i\sqrt{\frac{11}{4}-\frac{e^{2} }{3} }[/tex]
