What is the formula that gives the x-coordinate of the vertex for a parabola?

the formula is x = -b/(2a), where the parabola is of the form y = ax^2 + bx + c.

we can derive this by completing the square

y = ax^2 + bx + c

= a(x^2 + (b/a)x   ) + c

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

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

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

or by using calculus which uses the fact that that , so the critical point (always a global maximum or minimum in the case of a parabola) occurs at x = -b/(2a).