Divide through out by 3 giving:

#y=x^2-7/3x-2/3#

British name for this is: completing the square

You transform this into a perfect square with inbuilt correction as follows:

#color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

#color(brown)("Consider the part that is: "x^2-7/3x)#

#color(brown)("Take the"(-7/3)"and halve it. So we have"1/2 xx(-7/3)=(-7/6))#

#color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

Now write: #y-> (x-7/6)^2-2/3#

I have not used the equals sign because an error has been introduced. Once that error is removed we can then start to use the = sign again.

#color(white)(xxxxxxxx)"----------------------------------------------"#

#color(red)(underline("Finding the introduced error"))#

If we expand the brackets we get:

#color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2)-2/3#

The blue is the error.

#color(white)(xxxxxxxx)"----------------------------------------------"#

#color(red)(underline("Correction for the introduced error"))#

We correct for this by subtracting the same value so that we have:

#color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2-(7/6)^2)-2/3#

Now lets change the bit in green back to where it came from:

#color(green)(y->x^2- 7/3 x+(7/6)^2color(blue)(-(7/6)^2-2/3))#

Giving:

#color(green)(y= (x-7/6)^2)color(blue)(-(7/6)^2-2/3#

The equals sign (=) is now back as I have included the correction.

#color(white)(xxxxxxxx)"----------------------------------------------"#

#color(red)(underline("Finalising the calculation"))#

Now we can write:

#y= (x-7/6)^2-(49/36)-2/3#

#2 1/36#

#color(green)(y= (x-7/6)^2-73/36)#