We have that
[tex]-3x^2 + x + 3 = 9[/tex].
First we should transform it so that the left hand side is equal to 0, and the [tex]x^2[/tex] coefficient (that is, the number of [tex]x^2[/tex]s you have) is equal to 1.
So first we will subtract 9, giving
[tex]-3x^2 + x -6 = 0[/tex].
Now we will multiply by -1, giving
[tex]3x^2 - x + 6 = 0[/tex],
and finally we will divide by 3 to make the [tex]x^2[/tex] coefficient equal to 1, giving
[tex]x^2 - \frac{1}{3}x + 2 = 0[/tex].
Now it is the form we need to complete the square. Notice that expanding [tex](x+p)^2[/tex] gives
[tex]x^2 + 2px + p^2.[/tex]
Now, we have [tex]\frac{1}{3} x[/tex] in our equation so we want
[tex]2p = \frac{1}{3}
\implies p = \frac{1}{6}[/tex].
If [tex]p = \frac{1}{6}[/tex] then, from our expansion above, we have that
[tex](x+\frac{1}{6})^2 = x^2 + \frac{1}{3}x + \frac{1}{36}[/tex],
which is almost our equation, but not quite. Instead of [tex]\frac{1}{36}[/tex], we have 2. So we need to add [tex]\frac{71}{36}[/tex], giving:
[tex](x+\frac{1}{6})^2 + \frac{71}{36} = x^2 + \frac{1}{3}x + 2 = 0[/tex].
So, finally, subtracting the [tex]\frac{71}{36}[/tex], we have
[tex](x+\frac{1}{6})^2 = -\frac{71}{36}[/tex]
which is in the required form.
