whee
remember
[tex](x^m)(x^n)=x^{m+n}[/tex]
multiply both sides by 2^x
[tex](2^{2x})+(2^0)=3(2^x)[/tex]
[tex]2^{2x}+1=3(2^x)[/tex]
minus 3(2)^x from both sides
[tex]2^{2x}-3(2^x)+1=0[/tex]
we can use the quadratic formula
use u subsitution where u=2ˣ
[tex]1(2^x)^2-3(2^x)+1=0[/tex]
[tex]1(u)^2-3(u)+1=0[/tex]
[tex]u^2-3u+1=0[/tex]
use quadratic formula
[tex]u=\frac{3+\sqrt{5}}{2}[/tex] and [tex]u=\frac{3-\sqrt{5}}{2}[/tex]
solve
subsitte and solve
u=2ˣ
[tex]2^x=\frac{3+\sqrt{5}}{2}[/tex]
take ln of both sides
[tex]x(ln(2))=ln(\frac{3+\sqrt{5}}{2})[/tex]
divide both sides by ln(2)
[tex]x=\frac{ln(\frac{3+\sqrt{5}}{2})}{ln(2)}[/tex]
for other one
[tex]2^x=\frac{3-\sqrt{5}}{2}[/tex]
take ln of both sides
[tex]x(ln(2))=ln(\frac{3-\sqrt{5}}{2})[/tex]
divide both sides by ln(2)
[tex]x=\frac{ln(\frac{3-\sqrt{5}}{2})}{ln(2)}[/tex]
so [tex]x=\frac{ln(\frac{3+\sqrt{5}}{2})}{ln(2)}[/tex] and [tex]x=\frac{ln(\frac{3-\sqrt{5}}{2})}{ln(2)}[/tex]
those are the 2 solutions
aprox x=1.38848 and x=-1.338848
