your equation is (2x - 1) ^ 2 = - 4
simplify to get 4x^2 -4x + 1 = -4
add 4 to both sides of the equation to get:
4x^2 - 4x + 5 = 0
in this standad form:
a =
b = -4
c = 5
your solution is that c = 5.
the two equations of:
(2x-1)^2 = -4
4x^2 - 4x + 5 = 0
are equivalent.
to show this on a graph, let y = (2x-1)^2 + 4 and let y = 4x^2 - 4x + 5, and graph them on the same graph.
the graphs will be identical, showing only one graphed equation, rather than two separate graphed equations.
that says that the graph are identical.
that means the equations are just different forms of the same equation.
the minimum point on the graph for both equations is (.5,4)
when x = .5:
y = 4x^2 - 4x + 4 = 4 * .5^2 - 4 * .5 + 5 = 1 - 2 + 5 = 4
y = (2x - 1)^2 + 4 = (2 * .5 - 1)^2 + 4 = 0^2 + 4 = 4.
both equations give the same value.
when they do this for all possible values of x, the equations are equivalent.
the graph looks like this:
i took the time to show you this in an excel spreadsheet, as shown below:
as you can see, all values of y are the same for the same value of x.