Explanation needed for formula x=-b/2a

[Guest]

hi, I know that to find the minimum point of a graph [tex]y=ax^2+bx+c[/tex] we use the formula [tex]x=-b/2a[/tex], but I don't understand why this formula works. Can someone give an explanation? Thank you!

[HallsofIvy]

Compete the square!

[tex]ax^2+ bx+ c= a(x^2+ (b/a)x)+ c[/tex]

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

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

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

A square is never negative so (as long as a is positive which has to be true though you did not say it) [tex]a(x+ b/2a)^2[/tex] is non-negative. The smallest value of [tex]a(x+ b/2a)^2[/tex] is 0 which happens when x+ b/2a= 0 or x= -b/2a when it has value [tex]c- b^2/4a[/tex]. For any other value of x, it is [tex]c- b^2/4a[/tex] plus some positive value.

[Guest]

Thank you!