Why is the vertex formula -b/2a, - discriminent/4a. What the meaning being it

Why is the vertex formula -b/2a, - discriminent/4a. What the meaning being it?

The discriminant is just a part of the factored version of ax^2 + bx + c = 0 used in the zero formula where the x is isolated. But why it the x value of the vertex -b/2a and y value -discriminant/4a.

What the meaning behind, is it another factored version of ax^2 + bx + c = 0?

There is a close relationship between the things that you are looking at.
One thing you are looking at is the equation ax^2 + bx + c = 0. The discriminant tells you about the nature of the roots of that equation.
Another thing you are looking at is the parabola ax^2 + bx + c = y

I think you are treating these as separate and distinct objects, when actually you want a way to unify them, Your instincts are telling you that they should be related since the discriminant is involved in both of them, but your question is really, how are they related?

What you need to look at is the function f(x) = ax^2 + bx + c. The parabola is just the graph of that equation. And the zeroes of that function are just the solution to the equation.

Let's derive the formulas for the vertex given in the question. Note how similar this is to the derivation of the quadratic formula giving you the solutions of the equation.

Start with ax^2 + bx + c = y

To find the axis of symmetry, you need to complete the square
x^2 + b/a x = y/a - c/a
x^2 + b/a x + b^2/4a^2 = (x+b/2a)^2 = y/a - c/a + b^2/4a^2
So the line of symmetry turns out to be x = -b/2a. The vertex will lie on that line.

To find the y coordinate of the vertex, plug in -b/2a for x, and you get
0 = y/a - c/a + b^2/4a^2
Multiply by a:
0 = y - c + b^2/4a
y = (- b^2 +4ac)/4a = - discriminant / 4a

How can we relate that to solving ax^2 +bx + c = 0 and how the discriminant of that equation gives us information?

Let's first make the assumption, WLOG, that a > 0, so that we have a vertical parabola opening upwards.