Multiply the brackets giving:
#y=x^2-12x+8x-96" "->" "y=x^2-4x-96#.......Equation(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 1")#
Write as:
#y=(x^2-4x)-96+k#
where #k# is a correction that cancels out the error produced by the following process.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#
Take the power from #x^2# and move it outside the brackets
#y=(x-4x)^2-96+k#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
Discard the #x# from #4x#
#y=(x-4)^2-96+k#
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
halve the #-4# inside the brackets
#y=(x-2)^2-96+k" "larr" Now we need the "k#...Equation(2)
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4")#
If you were to expand the bracket you would have:
#y=x^2-4xcolor(magenta)(+4)-96+k#
The #color(magenta)(+4)# is the error. If you compare this to Equation(1) you will find that the +4 is not in it
So we set #+4+k=0 => k=-4#
So Equation(2) becomes:
#color(brown)(y=(x-2)^2-96+kcolor(blue)(" "->" "y=(x-2)^2-96-4)#
#color(green)(y=(x-2)^2-100#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(magenta)("General case")#
#y= ax^2+bx+c " "->" "y=a(x+b/(2a))^2+c -[a(b/(2a))^2]#
#" "uarr#
#" "k#
