Find the value of x that satisfies the equation:
2^(2x) - 2^(x + 1) + 15 = 0
Hint: Try making a substitution to simplify the equation before solving for x.
Solution:
Let y = 2^x
Then the equation can be rewritten as:
y^2 - 2y + 15 = 0
Using the quadratic formula:
y = (2 ± sqrt(4 - 4*15)) / 2
y = 1 ± 2i√2
Since y = 2^x, we have:
2^x = 1 ± 2i√2
Taking the logarithm of both sides:
x log(2) = log(1 ± 2i√2)
Using the fact that log(a ± b) ≠ log(a) ± log(b):
x log(2) = log(√(1 + 8) ± i√(8 - 1))
Using the fact that log(sqrt(a)) = (1/2)log(a):
x log(2) = (1/2)log(9) ± (1/2)log(7)
x = (1/2)log(9)/log(2) ± (1/2)log(7)/log(2)
x ≈ 1.77 ± 0.69i