To solve this equation, we need to isolate the square root term on one side of the equation. Here's how:

√( x^2-3x+1) = 4x-1

Square both sides of the equation to eliminate the square root:

(x^2-3x+1) = (4x-1)^2

Expand the right side:

x^2 - 3x + 1 = 16x^2 - 8x + 1

Rearrange the equation to bring all terms to one side:

16x^2 - 9x^2 - 8x + 3x = 0

Combine like terms:

7x^2 - 5x = 0

Factor out an x:

x(7x - 5) = 0

Set each factor equal to zero and solve for x:

x = 0 or 7x - 5 = 0

If x = 0, substitute it back into the original equation to check if it satisfies the equation:

√(0^2-3(0)+1) = 4(0)-1
√(0) = -1
0 = -1 (Not true)

Now solve for x when 7x - 5 = 0:

7x = 5
x = 5/7

So the solution to the equation is x = 5/7.