Very simple. Adjust the denominator 4^x -> 2^(2x). Then use the rule (a^x)/(a^y)=a^(x-y) to transform the equation to 2^(x^2-2*x)=8=2^3. Since the bases are the same, the exponents must be equal, so x^2-2*x=3, a quadratic equation with roots 3 and -1. Even easier than factoring is adding 1 to each side of the quadratic to get x^2-2*x+1=4, or (x-1)^2=4, so x-1=+-2.
The verification makes too much use of the same algebraic transformations as were used in the solving. If there were a mistake you would likely be repeating it. Just crunch the numbers.
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Very simple. Adjust the denominator 4^x -> 2^(2x). Then use the rule (a^x)/(a^y)=a^(x-y) to transform the equation to 2^(x^2-2*x)=8=2^3. Since the bases are the same, the exponents must be equal, so x^2-2*x=3, a quadratic equation with roots 3 and -1. Even easier than factoring is adding 1 to each side of the quadratic to get x^2-2*x+1=4, or (x-1)^2=4, so x-1=+-2.
Noted. Thank you for watching.
All fine until needlessly complicated factoring of quadratic equation: x^2 -2x -3=0 which is simply (x+1)(x-3)=0; x= -1 or +3.
Noted. Thanks for watching.
The verification makes too much use of the same algebraic transformations as were used in the solving. If there were a mistake you would likely be repeating it. Just crunch the numbers.
Noted.
2^(x^2)=2^3 * 2^2x , 2^(x^2)=2^(2x+3) , x^2=2x+3 , x^2-2x-3=0 , x= 3 , -1 ,
You are correct.