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)#
