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)#