Why is the discriminant formula b squared - 4ac, why b square? Why -4ac

I understand that it to find the zeros but why is it b square, why is it - 4ac. I care about the meaning of why the part of the formula are the way they are.

ax^2 + bx + c = 0

by completing the square

x^2 + bx/a + c/a = 0

x^2 + bx/a = -c/a

x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2

(x + b/2a)^2 = -c/a + b^2/4a^2

(x + b/2a)^2 = (b^2 - 4ac) / 4a^2

x + b/2a = ± √[(b^2 - 4ac) / 4a^2]

x + b/2a = ± √[(b^2 - 4ac) / 2a

x = [-b ± √(b^2 - 4ac)] / 2a


the discriminant is
b^2 - 4ac

if
b^2 - 4ac > 0, there are two real solutions
b^2 - 4ac < 0, there are two complex solutions
b^2 - 4ac = 0, there is one solution (multiplicity of two)

also
if b^2 - 4ac >0 AND is a perfect square.....you can factor it with integers (rational solutions)

The formula for solving a quadratic equation is obtained by using the completing the square method on the general quadratic equation ax² + bx + c = 0.

This naturally throws up a discriminant of b² - 4ac.

I could show you this if you wish.