A quadratic is written in the form #y= ax^2 +bx+c#
Vertex form is known as #y = a(x+b)^2 +c,# giving the vertex as #(-b,c)#
It is useful to be able to change a quadratic expression into the form #a(x+b)^2 +c#. The process is by completing the square.
#y =9x^2 +14x+12" "larr# the coefficient of #x^2# must be #1#
#y =9(x^2 +14/9x +12/9)#
To make a square of a binomial, you need to add on #color(blue)((b/2)^2)#
It is also subtracted so that the value of the expression is not changed. #color(blue)((b/2)^2 -(b/2)^2=0)#
#y =9(x^2 +14/9x color(blue)(+ (7/9)^2 -(7/9)^2) +12/9)#
#y = 9(color(red)((x^2 +14/9x + (7/9)^2))+color(green)(( -49/81 +12/9)))#
#y= 9(color(red)((x+7/9)^2 +color(green)((-49/81 12/9))))#
#y=9(x+7/9)^2+9(-49/81+108/81)#
#y = 9(x+7/9)^2 + 9(59/108))#
#y = 9(x+7/9)^2 +59/12#
