#y=(3x-1)(2x+11)#
Multiply the brackets
#y=6x^2+33x-2x-11#
#y=6x^2+31x-11 larr" Starting point"#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Discussing what is happening")#
Note that for standardised form #y=ax^2+bx+c# we intend to make this #y=a(x+b/(2a))^2+k+c color(white)(.) larr" completed square format"#
If you multiply out the whole thing we get:
#y=ax^2+b x color(red)(+ a( b/(2a))^2)+k+c#
The #color(red)(+ a( b/(2a))^2)+k# is not in the original equation.
To 'force' this back to the original equation we
set #color(red)(+ a( b/(2a))^2)+k=0#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Returning to the solution")#
#y=6x^2+31x-11 color(white)("d")->color(white)("d")y=6(x+31/(6xx2))^2 +k-11#
However:
#color(red)(+ a( b/(2a))^2)+k=0 color(white)("d")->color(white)("dddd") color(red)(6(31/(2xx6))^2)+k=0#
#color(white)("dddddddddddddddd")->color(white)("dddd")31^2/(4xx6)+k=0#
#color(white)("dddddddddddddddd")->color(white)("dddd") k=-961/24#
So we now have:
#y=6x^2+31x-11 color(white)("d")->color(white)("ddd")y=6(x+31/(6xx2))^2 -1225/24#
#color(white)("dddddddddddddddd")->color(white)("dddd") y=6(x+31/12)^2-1225/24#
