4x^{2} + 4y^{2} - 5x = 0 (1)

The equation of the circle is in the general form:

x^{2} + y^{2} + 2gx + 2fy + c = 0

We want to write equation in terms of its center and radius:

(x - h)^{2} + (y - k)^{2} = r^{2} (2)

4x^{2} + 4y^{2} - 5x = 0

Collect like terms:

(4x^{2} - 5x) + 4y^{2} = 0

Divide both sides by 4, to make the coefficients of x^{2} and y^{2} to be unity (i.e. 1):

x^{2} - (5/4)x + y^{2} = 0

Complete the square on both sides (i.e. add half the square of the coefficient of x on both sides of the equation):

x^{2} - (5/4)x + (-5/4*1/2)^{2} + y^{2} = (-5/4*1/2)^{2}

x^{2} - (5/4)x + (-5/8)^{2} + y^{2} = (-5/8)^{2}

Rewrite the coefficient of x as twice the square constant:

x^{2} - 2(5/8)x + (-5/8)^{2} + y^{2} = (-5/8)^{2}

Factoring gives:

(x - 5/8)(x - 5/8) + y^{2} = (-5/8)^{2}

(x - 5/8)^{2} + y^{2} = 25/64

Comparing with equation (2): (x - h)^{2} + (y - k)^{2} = r^{2}

h = 5/8, k = 0, r = √(25/64) = 5/8

Center: **(h, k) = (5/8, 0)**

Radius: **r = 5/8**

**Check:**

(x - 5/8)^{2} + y^{2} = 25/64

Expand:

x^{2} - 2(5/8)x + (-5/8)^{2} + y^{2} = 25/64

x^{2} - 2(5/8)x + 25/64 + y^{2} = 25/64

x^{2} - (5/4)x + y^{2} = 0

Multiply by 4 throughout:

**4x**^{2}** + 4y**^{2}** - 5x = 0**