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)
