Solve for x over the real numbers:
-27+3^(2 x)-2 3^(x+1) = 0
Simplify and substitute y = 3^x:
-27+3^(2 x)-2 3^(x+1) = -27-6×3^x+(3^x)^2 = y^2-6 y-27 = 0:
y^2-6 y-27 = 0
The left hand side factors into a product with two terms:
(y-9) (y+3) = 0
Split into two equations:
y-9 = 0 or y+3 = 0
Add 9 to both sides:
y = 9 or y+3 = 0
Substitute back for y = 3^x:
3^x = 9 or y+3 = 0
9 = 3^2:
3^x = 3^2 or y+3 = 0
Equate exponents of 3 on both sides:
x = 2 or y+3 = 0
Subtract 3 from both sides:
x = 2 or y = -3
Substitute back for y = 3^x:
x = 2 or 3^x = -3
3^x = -3 has no solution since for all z element R, 3^z>0 and -3<0:
Answer: |x = 2
