Solve for x over the real numbers:
3^(2 x)-3^(2-2 x) = 8
Simplify and substitute y = 3^(2 x):
3^(2 x)-3^(2-2 x) = 3^(2 x)-(9)/3^(2 x) = y-9/y = 8:
y-9/y = 8
Bring y-9/y together using the common denominator y:
(y^2-9)/y = 8
Multiply both sides by y:
y^2-9 = 8 y
Subtract 8 y from both sides:
y^2-8 y-9 = 0
The left hand side factors into a product with two terms:
(y-9) (y+1) = 0
Split into two equations:
y-9 = 0 or y+1 = 0
Add 9 to both sides:
y = 9 or y+1 = 0
Substitute back for y = 3^(2 x):
3^(2 x) = 9 or y+1 = 0
9 = 3^2:
3^(2 x) = 3^2 or y+1 = 0
Equate exponents of 3 on both sides:
2 x = 2 or y+1 = 0
All equations give x = 1 as the solution:
x = 1 or y+1 = 0
Subtract 1 from both sides:
x = 1 or y = -1
Substitute back for y = 3^(2 x):
x = 1 or 3^(2 x) = -1
3^(2 x) = -1 has no solution since for all z element R, 3^z>0 and -1<0:
Answer: | x = 1
COMPLEX SOLUTION:
Solve for x:
3^(2 x)-3^(2-2 x) = 8
Simplify and substitute y = 3^(2 x):
3^(2 x)-3^(2-2 x) = 3^(2 x)-(9)/3^(2 x) = y-9/y = 8:
y-9/y = 8
Bring y-9/y together using the common denominator y:
(y^2-9)/y = 8
Multiply both sides by y:
y^2-9 = 8 y
Subtract 8 y from both sides:
y^2-8 y-9 = 0
The left hand side factors into a product with two terms:
(y-9) (y+1) = 0
Split into two equations:
y-9 = 0 or y+1 = 0
Add 9 to both sides:
y = 9 or y+1 = 0
Substitute back for y = 3^(2 x):
3^(2 x) = 9 or y+1 = 0
Take the logarithm base 3 of both sides:
2 x = 2+(2 i π n_1)/(log(3)) for n_1 element Z
or y+1 = 0
Divide both sides by 2:
x = 1+(i π n_1)/(log(3)) for n_1 element Z
or y+1 = 0
Subtract 1 from both sides:
x = 1+(i π n_1)/(log(3)) for n_1 element Z
or y = -1
Substitute back for y = 3^(2 x):
x = 1+(i π n_1)/(log(3)) for n_1 element Z
or 3^(2 x) = -1
Take the logarithm base 3 of both sides:
x = 1+(i π n_1)/(log(3)) for n_1 element Z
or 2 x = (i π (1+2 n_2))/(log(3)) for n_2 element Z
Divide both sides by 2:
Answer: | x = 1+(i π n_1)/(log(3)) for n_1 element Z
or x = (i π (1+2 n_2))/(2 log(3)) for n_2 element Z
