#color(blue)("Determine the structure of the vertex form")#
Multiply out the brackets giving:
#y=2x^2+10x+13x+65#
#y=2x^2+23x+65" "#...................................(1)
write as:
#y=2(x^2+23/2x)+65#
What we are about to do will introduce an error for the constant. We get round this by introducing a correction.
Let the correction be k then we have
#color(brown)(y=2(x+23/4)^2+k+65" ")#..................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~
To get to this point I moved the square from #x^2# to outside the brackets. I also multiplied the coefficient of #23/2x# by #1/2# giving the #23/4# inside the brackets.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the value of the correction")#
We need the values of a point for substitution so that k can be calculated.
Using equation (1) set #x=0# giving
#y=2(0)^2+23(0)+65 => y=65#
So we have our ordered pair of #(x,y)->(0,65)#
Substitute this into equation (2) giving:
#cancel(65)=2(0+23/4)^2+k+cancel(65)" ".................................(2_a)#
#k=-529/8#
#y=2(x+23/4)^2-529/8+65" "#..................................(3)
But#" "65-529/8 = 9/8#
Substitute into equation (3) gives:
#color(blue)("vertex form "->" "y=2(x+23/4)^2+9/8)#
#color(brown)("Note that "(-1)xx23/4 = -5 3/4 ->" axis if symmetry")#
