Your equation has complex solutions. They can be found by "completing the square" or by using the quadratic formula.
4x^2 - 4x + 3 = 0
4x^2 - 4x = -3 (removing the constant from the left side can make it easier to see how to complete the square)
4x^2 - 4x + 1 = -2 (add (b^2)/(4a) to make ax^2 + bx into a perfect square)
(2x - 1)^2 = -2 (rewrite as a square)
2x - 1 = ±√(-2) = ±I√2 (take the square root of both sides)
2x = 1 ± I√2 (add 1)
x = (1 ± I√2)/2 (divide by 2)
4x^2 - 4x + 3 = 0
4x^2 - 4x = -3 (removing the constant from the left side can make it easier to see how to complete the square)
4x^2 - 4x + 1 = -2 (add (b^2)/(4a) to make ax^2 + bx into a perfect square)
(2x - 1)^2 = -2 (rewrite as a square)
2x - 1 = ±√(-2) = ±I√2 (take the square root of both sides)
2x = 1 ± I√2 (add 1)
x = (1 ± I√2)/2 (divide by 2)
