Solving Exponential Equation @KasyannoEZMath

This tutorial video shows how to solve the given exponential equation.

[gelbkehlchen]

Solution:
27^(x²)-9^(x²) = 3^(x²) ⟹
3^(3*x²)-3^(2*x²) = 3^(x²) ⟹
[3^(x²)]³-[3^(x²)]² = 3^(x²) |with u=3^(x²) ⟹
u³-u² = u |-u ⟹
u³-u²-u = 0 ⟹
u*(u²-u-1) = 0 |either u1=0 or u²-u-1 = 0 |p-q-formula ⟹
u2/3 = 1/2±√(1/4+1) = 1/2±1/2*√5 = (1±√5)/2 ⟹
u2 = (1+√5)/2 and u3 = (1-√5)/2 ⟹

1.case: 3^(x1²) = u1 = 0 |that is not defined.

2.case: 3^(x2²) = u2 = (1+√5)/2 |ln[ ] ⟹
x2²*ln(3) = ln[(1+√5)/2] |/ln(3) ⟹
x2² = ln[(1+√5)/2]/ln(3) |√{ } ⟹
x21/2 = ±√{ ln[(1+√5)/2]/ln(3)} ⟹
x21 = +√{ln[(1+√5)/2]/ln(3)} = +0.6618291920774895 and
x22 = -√{ln[(1+√5)/2]/ln(3)} = -0.6618291920774895

3.case: 3^(x3²) = u3 = (1-√5)/2 < 0 |that is not defined.

sample:
left side: 1.6180339887498957
right side: 1.6180339887498951 all o.k.