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Find all x so that 8^(x^2)*4^x=2. Enter all the solutions, separated by commas.

 May 5, 2022
 #1
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8^(x^2)*4^x = 2

 

We can write

 

(2^3)^(x^2)*(2^2)^x  = 2^1

 

[2^(3x^2) ] * [ 2^(2x) ]  =  2^1

 

2^ [ (3x^2 + 2x) ]  = 2^1             we have the bases equal....solve for the exponents

 

3x^2 + 2x  =  1      complete the square

 

3(x^2 + (2/3)x + 1/9)^2  = 1  + 1/3

 

3 ( x + 1/3)^2  = 4/3                      multiply both sides by 1/3

 

(x + 1/3)^2  =  4/9              take both roots

 

x + 1/3 =   2/3                or        x + 1/3  =  -2/3

 

x   = 2/3 - 1/3                            x = -2/3 - 1/3

 

x = 1/3                                      x =  -1

 

cool cool cool

 May 6, 2022

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