Expand the expression in the equation
$$\left(2 x - 1\right)^{2} = 0$$
We get the quadratic equation
$$4 x^{2} - 4 x + 1 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = -4$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (4) * (1) = 0
Because D = 0, then the equation has one root.
x = -b/2a = --4/2/(4)
$$x_{1} = \frac{1}{2}$$