The general vertex form is

#color(white)("XXX")y=(x-color(red)a)^2+color(blue)b#

with vertex at #(color(red)a,color(blue)b#)

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| The process used (below) is often referred to as#color(white)("XXXxx")# |

| #color(white)("XXX")#*completing the square*#color(white)("XXXXXXXXXXXXXX..x")#|

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In order to convert the given equation: #y=x^2-4x+3#

into vertex form, we need to a term of the form #(x-color(red)a)^2#

Since #(x-color(red)a)^2=(x^2+2color(red)ax+color(red)a^2)#

and

since the coefficient of #x# in the given equation is #(-4)#

then

#color(white)("XXX")2color(red)a=-4color(white)("xxx")rarrcolor(white)("xxx")color(red)a=-2#

and

#color(white)("XXXXXXXXXXXXXXX")color(red)a^2=color(green)4#

That is the first term of the vertex form must be

#color(white)("XXX")(x-color(red)2)^2#

We need to add #color(green)4# to #x^2-4x# from the original equation to make #(x-color(red)2)^2#

...but if we are going to **add** #color(green)4# then we will also need to **subtract** #color(green)4# so the equation will not really change.

#y=x^2-4x+3#

#color(white)("XXX")#will therefore become

#y=underbrace(x^2-4xcolor(green)(+4))+underbrace(3color(green)(-4))#

#color(white)("XXX")#re-writing as a squared binomial and simplifying the constants

#y=underbrace((x-color(red)2)^2)+underbrace(color(blue)((-1)))#

#color(white)("XXX")#which is the vertex form