Solving Exponentials - Al 2 - EASY BEST ANSWER!!?

Question

I can't solve these for the life of me - whoever does correctly will get best answer!! Pleaase help - I want to understand:)

Solve 9^(2x^2 - 2x) = 27

Solve 8(3^x - 6) = 876

Answer 1

9^(2x^2 - 2x) = 27
3^(2(2x^2 - 2x)) = 3^3
2(2x^2 - 2x) = 3
4x^2 - 4x - 3 = 0
(2x + 1)(2x - 3) = 0
x = -1/2 or 3/2

8(3^x - 6) = 876
3^x - 6 = 219/2
3^x = 231/2
ln 3^x = ln 231/2
x ln 3 = ln 231/2
x = (ln 231/2) / (ln 3)
x = 4.323

Answer 2

9^(2x^2-2x)=27
The trick with these things is to get similar bases. If you can do that, then you can equate the exponents.
9^(2x^2-2x)=9^(3/2)
2x^2-2x=3/2. Has to be. The bases are the same.
4x^2-4x=3
4x^2-4x-3=0
(2x-3)(2x+1)=0
2x-3=0, and so 2x=3, x=3/2
or 2x+1=0, 2x=-1, x=-1/2
Substituting x=3/2 and x=-1/2 back into the original
equation confirms the equality, 9^(3/2)=9^(3/2), so
the answers are correct.

8(3^x-6)=876
This one confuses me. As written, I read this as
8 times 3 exponent x, minus 6, the 6 NOT being part
os the exponent. In other words, you did NOT write
this as 8^(3x-6).
If I am correct in my assumption,
8(3^x-6)=876
3^x-6=109.5
3^x=109.5+6
3^x=115.5
xlog3=log115.5
x=log115.5 /log3
I don't have log tables,so I must leave it to you to complete.
Be wary of my work- my assumption may be way
off base(pun intended).

Answer 3

Solve 9^(2x^2 - 2x) = 27

9 = 3^2
27 = 3^3
3^(2(2x^2 - 2x)) = 3^3

take the ln of both sides to get
2(2x^2 -2x) ln(3) = 3 ln(3)
simplify by ln(3) to get
2(2x^2 -2x) = 3
4x^2 -4x -3 =0
D = 16 +48 = 64 = 8^2
x1 = (4 -8)/8 = -1/2
x2 = (4+8)/8 = 12/8 = 3/2
Summary:
Roots are:
x1 = -0.5
x2 = 1.5

**********************************
Solve 8(3^x - 6) = 876

3^x -6 = 876/8
3^x = 876/8 +6 =924/8 = 231/2
take the ln of both sides to get
x ln(3) = ln(231/2)
x = ln(231/2) /ln(3) or approximately
x = 4.32297

Answer 4

9^(2x² - 2x) = 27
(3²)^(2x² - 2x) = 3³
3^(4x² - 4x) = 3³
4x² - 4x = 3
4x² - 4x - 3 = 0
(2x - 3)(2x + 1) = 0
x = 3/2 or x = -1/2

8(3^x - 6) = 876
3^x - 6 = 109.5
3^x = 115.5
x log 3 = log 115.5
x = log 115.5 / log 3
x = 4.323