Answer: x = 5.2136 or x = -1.2136
Step-by-step explanation: Let's first simplify the expression by using the identity:
(a+b)^2 = a^2 + 2ab + b^2
For this problem, let a = sqrt(2) and b = 1, so we get:
(sqrt{2}+1)^2 = 2 + 2sqrt{2} + 1 = 3 + 2sqrt{2}
Now we can rewrite the original equation as:
(sqrt{2}+1)^x + (sqrt{2}-1)^x = 6
[(sqrt{2}+1)^(x/2)]^2 + [(sqrt{2}-1)^(x/2)]^2 + 2(sqrt{2}+1)^(x/2)(sqrt{2}-1)^(x/2) = 6
Let's define a = (sqrt{2}+1)^(x/2) and b = (sqrt{2}-1)^(x/2), then the equation can be rewritten as:
a^2 + b^2 + 2ab = 6
Now we can substitute the value we found earlier for (sqrt{2}+1)^2:
a^2 + b^2 + 2ab = 3 + 2sqrt{2}
We can rewrite this as:
(a+b)^2 = 3 + 2sqrt{2}
Taking the square root of both sides, we get:
a+b = ±√(3 + 2sqrt{2})
Now we can substitute back in the expressions for a and b:
(sqrt{2}+1)^(x/2) + (sqrt{2}-1)^(x/2) = ±√(3 + 2sqrt{2})
We can solve for x/2 by taking the logarithm of both sides:
log[(sqrt{2}+1)^(x/2) + (sqrt{2}-1)^(x/2)] = log[±√(3 + 2sqrt{2})]
x/2 = 2.6068 or -0.6068
Multiplying both sides by 2, we get:
x = 5.2136 or -1.2136
Therefore, the solutions are:
x = 5.2136 or x = -1.2136.