Help with this quadratic equation?

x² - 5x - 3 = 0
(x - 5/2)² - (5/2)² - 3 = 0
(x - 5/2)² - 9.25 = 0
(x - 5/2)² = 9.25
x - 5/2 = ±√9.25
x = ±√9.25 + 2.5
x = √9.25 + 2.5 or -√9.25 + 2.5

Question Number 1 :
For this equation x^2 - 5*x - 3 = 0 , answer the following questions :
A. Use completing the square to find the root of the equation !

Answer Number 1 :
The equation x^2 - 5*x - 3 = 0 is already in a*x^2+b*x+c=0 form.
As the value is already arranged in a*x^2+b*x+c=0 form, we get the value of a = 1, b = -5, c = -3.

1A. Use completing the square to find the root of the equation !
x^2 - 5*x - 3 = 0 ,divide both side with 1
Which result in x^2 - 5*x - 3 = 0 ,
And the coefficient of x is -5
We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = -5/2 = -2.5
So we have make the equation into x^2 - 5*x + 6.25 - 9.25 = 0
So we will get ( x - 2.5 )^2 - 9.25 = 0
And it is the same with (( x - 2.5 ) - 3.04138126514911 ) * (( x - 2.5 ) + 3.04138126514911 ) = 0
Which is the same with ( x - 2.5 - 3.04138126514911 ) * ( x - 2.5 + 3.04138126514911 ) = 0
So we just need to add up, and get ( x - 5.54138126514911 ) * ( x + 0.54138126514911 ) = 0
So we got the answers as x1 = 5.54138126514911 and x2 = -0.54138126514911

x^2 - 5x - 3 = 0
x^2 - 5x = 3
x^2 - 5x/2 - 5x/2 = 3
x^2 - 5x/2 - 5x/2 + 25/4 = 3 + 25/4
(x^2 - 5x/2) - (5x/2 - 25/4) = 12/4 + 25/4
x(x - 5/2) - 5/2(x - 5/2) = 37/4
(x - 5/2)(x - 5/2) = 37/4
(x - 5/2)^2 = 37/4
x - 5/2 = ±√(37/4)
x = 5/2 ±(√37)/2
x = (5 ±√37)/2

x ² - 5 x = 3

x ² - 5 x + 25/4 = 3 + 25/4

( x - 5/2 ) ² = 37/4

x - 5/2 = ± √37/2

x = 5/2 ± √37/2

x = (1/2) [ 5 ± √37 ]