#color(blue)"Shortcut method - by sight")#
Given# -> y=x^2-3x-28# .......................................(1)
#y=(x-3/2)^2-3/4-28#
#y=(x-3/2)^2-121/4#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(purple)("Fuller explanation")#
#color(blue)("Step 1")#
Write as#" " y=(x^2-3x)-28#
#color(brown)("Divide the brackets contents by "x". These means that the right")##color(brown)("hand side is no longer equal to "y)#
#y!=(x-3)-28#
#color(brown)("square the brackets")#
#y!=(x-3)^2-28#
#color(brown)("Halve the -3 from "(x-3))#
#y!=(x-3/2)^2-28#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#
#color(brown)("Changing the equation so that it does equal "y)#
Let a constant of correction be k then
#y=(x-3/2)^2-28 + k#...................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
#color(brown)("To find the value of k")#
#color(green)("As equation (1) and equation (2) both equal y we can equate them")# #color(green)("to each other through y")#
Equation (1) = y = Equation (2)
# x^2-3x-28" "=" "(x-3/2)^2-28+k#
# cancel(x^2)-cancel(3x)-cancel(28)" "=" "cancel(x^2)-cancel(3x)+9/4-cancel(28)+k#
#k=-9/4#......................................................(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4 - last move!")#
#color(brown)("Bringing it all together to give the final equation")#
Substitute equation (3) into equation (2)
#y=(x-3/2)^2-28 -9/4#.
But #-28-9/4 = -121/4# giving
#color(green)(y=(x-3/2)^2-121/4#.
