This is a link to a step by step guide to my shortcut approach. When applied properly it should only take about 4 to 5 lines all depending on the complexity of the question. https://socratic.org/s/aMg2gXQm
The objective is to have the format #y=a(x+b/(2a))^2+c+k#
Where #k# is a correction making #y=a(x+b/(2a))^2+c color(white)("d")# have the same overall values as #y=ax^2+bx+c#
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#color(blue)("Answering the question - the more formal approach")#
#color(brown)("This is one of those situations where you just have to")##color(brown)("remember the standard form steps")#
Lets multiply out the brackets
#y=-1/7( 6x-3)(x/3+5) " "..................Equation(1)#
# y=-1/7(2x^2+30x-x-15)#
#y=-1/7(2x^2+29x-15)#
Factor out the 2 from #2x^2#. We do not want any coefficient in front of the #x^2#
# y=-2/7(x^2+29/2x-15/2) #
Just for ease of reference set #g=x^2+29/2x-15/2# giving:
#y=-2/7g" "..........................Equation(1_a)#
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The x-intercepts are at #y=0# giving
#y_("x-intercept")=0=-2/7g#
Thus it must be true that for this condition #g=0# thus we have:
#g=0=x^2+29/2x-15/2#
Add #15/2# to both sides
#15/2=x^2color(red)(+29/2)x#
To make the right hand side into a perfect square we need to add #(1/2xxcolor(red)(29/2))^2->(29/4)^2# so add #841/16# to both sides giving:
#15/2+ 841/16color(white)("d")=color(white)("d")x^2+29/2x+841/16#
#15/2+ 841/16color(white)("d")=color(white)("d")(x+29/4)^2#
#961/16color(white)("d")=color(white)("d")(x+29/4)^2#
#g=0=(x+29/4)^2-961/16#
But from #Equation(1_a)" "y=-2/7g# giving
#y=0=-2/7[(x+29/4)^2-961/16]#
#color(magenta)("I will let you finish this off.")#
