Use the method of completing the square to transform the quadratic equation into the equation form (x + p)^2 = q.
3 + x - 3x^2 = 9

We have that 
.

First we should transform it so that the left hand side is equal to 0, and the  coefficient (that is, the number of s you have) is equal to 1.

So first we will subtract 9, giving
.

Now we will multiply by -1, giving
,

and finally we will divide by 3 to make the  coefficient equal to 1, giving
.

Now it is the form we need to complete the square. Notice that expanding  gives


Now, we have  in our equation so we want 
.

If  then, from our expansion above, we have that
,
which is almost our equation, but not quite. Instead of , we have 2. So we need to add , giving:

.

So, finally, subtracting the , we have



which is in the required form.

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