#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")#