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Wednesday, 2 December 2015

Solve the following equation for x: 2^{2x + 1} - 9 \times 2^x + 4 = 0

Solve the following equation for x: 
    2^{2x + 1}  -  9 \times 2^x  +  4  =  0

 SOLUTION:  This is a quadratic in 2^x
           2 (2^x)^2 - 9 (2^x) +4 = 0 
         Use a substitution u=2^x and express the equation in terms of u.

Therefore:
  2u^2 - 9u +4 = 0 

Factorise to get 
( 2u- 1 )( u - 4 ) = 0
2u - 1 = 0      or     u - 4 = 0 
2u  = 1         or     u = 4 
u = 1/2          or     u=2^2

2^x=2^{-1}    or     2^x=2^2
 

            \therefore x=-1, 2 

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