[tex]\displaystyle\\Answer:\ x=\sqrt{2}\ \ \ \ \ x=-\sqrt{2}\ \ \ \ \ x=\frac{ln5}{2}[/tex]
Step-by-step explanation:
[tex]5x^2-x^2e^{2x}+2e^{2x}=10[/tex]
[tex]Let:\ x^2=t \ \ \ \ \ e^{2x}=v\\Hence,\\5t-tv+2v^2=10\\5t-tv+2v^2-10=10-10\\5t-tv-10+2v^2=0\\t(5-v)-2(5-v)=0\\(5-v)(t-2)=0\\5-v=0\\5-v+v=0+v\\5=v\\5=e^{2x}[/tex]
Prologarithmize both parts of the equation:
[tex]ln5=lne^{2x}\\ln5=2x*lne\\ln5=2x*1\\ln5=2x\\[/tex]
Divide both parts of the equation by 2:
[tex]\displaystyle\\x=\frac{ln5}{2}[/tex]
[tex]t-2=0\\t-2+2=0+2\\t=2\\x^2=2\\x=\sqrt{2} \\x=-\sqrt{2}[/tex]
