To solve this quartic equation, we can let y = x^2. Then we have:

y^2 - 4y + 1 = 0

Solving this quadratic equation using the quadratic formula, we get:

y = (4 ± √(16 - 4))/2
y = (4 ± √12)/2
y = (4 ± 2√3)/2
y = 2 ± √3

Since y = x^2, we have two possible solutions for x:

x^2 = 2 + √3
x = ±√(2 + √3)

x^2 = 2 - √3
x = ±√(2 - √3)

Therefore, the solutions to the quartic equation x^4 - 4x^2 + 1 = 0 are:

x = √(2 + √3), x = -√(2 + √3), x = √(2 - √3), x = -√(2 - √3)