Given:#" "y=2x^2+9x-5#......................(1)
Write as:#" "y=2(x^(color(magenta)(2))+9/2x)-5+k#
Where #k# is a correction factor for an unfortunate consequence of what we are about to do.
Take the power of 2 from #x^2# and move it to outside the brackets
#" "y=2(x+9/2color(blue)(x))^(color(magenta)(2))-5+k#
'Get rid' of the #color(blue)(x)# from #9/2color(blue)(x)#
#" "y=2(x+9/2)^2-5+k#
Apply #(-1/2)xx9/2 = -9/4#
#" "y=2(x+9/4)^2-5+k# .....................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The error comes from the #9/4# being squared. It introduces an extra value that we not there before. By the way, do not forget to multiply it by the constant of 2 outside the bracket.
So the error is #2(9/4)^2#
Consequently it has to be the case that: #2(9/4)^2+k=0#
so we have # 2(81/16) +k=0#
#=> k= -81/8#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So equation (2) becomes
#" "y=2(x+9/4)^2-5-81/8# .....................................(2_a)
Giving:
#color(blue)(" "y= 2(x+9/4)^2-121/8)#
