Given: #" "y=8x^2-6x+128# ..........(1)
Write as #" "y=8(x^2-6/8x)+128#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Now we start to change things a step at a time.")#
#color(green)("Change the bracket so that this part becomes:")#
#8{x-(1/2 xx6/8)}^2 #
#color(green)("Now put back the constant giving:")#
#8{x-(1/2 xx6/8)}^2 +128#
#color(green)("But this change has introduced an error so we can not yet equate it")# #color(green)("to "y.)#
#y!= 8{x-(1/2 xx6/8)}^2 +128#
#color(green)("We fix that by adding another constant ( say k ) giving:")#
#y=8{x-(1/2 xx6/8)}^2 +128+k# .........................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("To find the value of "k)#
#color(green)("Equate (2) to (1) through "y)#
#8{x-(1/2 xx6/8)}^2 +128+k" " =" "8x^2-6x+128#
#8(x^2-3/8x-3/8x+9/64)+128+k" "=" "8x^2-6x+128#
#cancel(8x^2)-cancel(6x)+9/8+cancel(128)+k" "=" "cancel(8x^2)-cancel(6x)+cancel(128)#
#k=-9/8# ....................................(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (3) into (2)
#color(blue)(y_("vertex form")=8(x-3/8)^2+ 126 7/8#
Note#" " 3/8 = 0.375#
So
#color(blue)(" "x_("vertex") = (-1)xx(-3/8) = + 0.375)#
#color(blue)(" "y_("vertex")= 126 7/8 = 126.875#
