You can very easily go wrong on this one. There is a small detail that can easily be over looked.
Let k be a constant yet to be determined
Given:" "y=1/5x^2-3/7x-16.......(1)
color(blue)("Build the vertex form equation")
Write as:" "y=1/5(x^2-color(green)(15/7)x)-16..........(2)
color(brown)("Note that "15/7xx1/5 = 3/7)
Consider the 15/7 "from "15/7x
Apply 1/2xx15/7 = color(red)(15/14)
At this point the right hand side will not be equal to y. This will be corrected later
In (2) substitute color(red)(15/14)" for "color(green)(15/7)
1/5(x^2-color(red)(15/14)x)-16" "....................(2_a)
Remove the x from 15/14x
1/5(x^(color(magenta)(2))-15/14)-16
Take the power (index) of color(magenta)(2) outside the bracket
1/5(x-15/14)^(color(magenta)(2))-16" "color(brown)("Note that an error comes from the "15/14
color(brown)("This is still not yet equal to y")
Add the constant value of color(red)(k)
1/5(x-15/14)^(color(magenta)(2))-16+color(red)(k)
color(green)("Now it is equal to "y)
y=1/5(x-15/14)^2-16+color(red)(k).........(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("To determine the value of "k)
If we were to expand the bracket and multiply by the 1/5 we would have the extra value of 1/5xx(-15/14)^2. The constant k is to counter this by removing it.
color(brown)("Let me show you what I mean. Compare equation (1) to (3)")
1/5x^2-3/7x-16" " =" "y" "=" "1/5(x-15/14)^2-16+k
1/5x^2-3/7x-16" " =" "1/5(x^2-15/7x+(15/14)^2)-16+k
1/5x^2-3/7x-16" " =1/5x^2-3/7x+[1/5xx(15/14)^2]-16+k
1/5x^2-3/7x-16" " =1/5x^2-3/7x+[45/196]-16+k
cancel(1/5x^2)-cancel(3/7x)-cancel(16)" " =cancel(1/5x^2)-cancel(3/7x)+[45/196]-cancel(16)+k
=>0=45/196+k
=>color(red)(k=-45/196)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So equation (3) becomes:
y=1/5(x-15/14)^2-16color(red)(-45/196).........(3)
y=1/5(x-15/14)^2-3181/196
color(blue)("Thus vertex form" -> y=1/5(x-15/14)^2-3181/196)