How do I factor this:
e^(2x)/144 + 9e^(-2x) + 1/2. Is there a method I can use that will work every time?

Question

How do I factor this:
e^(2x)/144 + 9e^(-2x) + 1/2.   Is there a method I can use that will work every time?

anonymous · Answer

@Preetha

anonymous · Answer

@Agent_Sniffles @agent @jiteshmeghwal9  @tkhunny

anonymous · Answer

well first we know that e^(-2x) is in the form 1/e^(2x)
so try to form an expantion$$(x+\frac{1}{x})^2$$

anonymous · Answer

$$(\frac{e^{2x}}{12}+\frac {3}{e^{2x}})^2$$

anonymous · Answer

notice that by inspection e^2x is no longer a problem but the numbers 12^2=144 and 3^9

anonymous · Answer

I'm trying to factor it so I can take the square root of the equation.

anonymous · Answer

the easy thing about the equation is that we have 
the form$$(x+\frac{1}{x})^2=x^2+2x \times \frac{1}{x}+\frac{1}{x^2}=x^2+\frac{1}{x^2}+2$$ so  we let 
$$e^{2x}=k$$
so
$$\frac{k^2}{144}+\frac{9}{k^2}+\frac{1}{2}$$
and this is 
$$(\frac{k}{12}+\frac{3}{k})(\frac{k}{12}+\frac{3}{k})=(\frac{k}{12}+\frac{3}{k})^2$$
putting back your k=e^2x
you get
 the factor