#color(blue)("Step 1")#
Write as:
#y=2(x^2+6x)+12+k#
When we start to change things the equation becomes untrue. So we need to introduce the correction #k# to compensate for this. The value of #k# is calculated at the end.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#
Move the power of 2 from #x^2# to outside the brackets.
#y=2(x+6x)^2+12+k#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
Halve the 6 from #6x#
#y=2(x+3x)^2+12+k#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
Remove the #x# from #3x#
#y=2(x+3)^2+12+k# .....................Equation(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4")#
Determine the value of #k#
If you were to square the bracket we would have #color(magenta)(3^2)# form #color(green)(2)(x+ color(magenta)(3))^2#. Also this is multiplied by the #color(green)(2)# from outside the bracket giving:
#color(green)(2xx)color(magenta)(3^2) larr" this is the error we introduced"#
So we write:
#color(green)(2xx)color(magenta)(3^2)+k=0#
#=> 18+k=0 -> k=-18#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 5 - The final equation")#
Substitute this into equation(1)
#y=2(x+3)^2+12-18#
#y=2(x+3)^2-6#
#color(red)("I have superimposed both graphs so that you can see they are the same.")#
