Answer:
Hi there! I don't really understand what you are trying to ask but when I looked at your working for completing the square for x²-2x+3, it looks correct.
Instead of memorizing formulas, I suggest that you understand the process of completing the square.
Below is the steps for completing the squares.
1. Ensure that the coefficient of x² is 1 ✓
2. Notice that the coefficient of the x term here is negative, so what we are trying to achieve is a² -2ab +b², which you could convert to (a-b)² later.
3. Find the values of a and b.
Since your coefficient of x² is a 1, the value of a in a²-2ab+b² is x.
➣x² -2x+3= x² -2(x) ....
4. Now find the value of b in a²-2ab+b².
By observation, you would see that b=1 since when you focus at the x term,
-2x= -2ab
-2x = -2(x)(b)
-2x = -2x (b)
b= 1
➣ So, x²-2x+3= x²-2(x)(1) +1²...
Now here you notice that the constant term is not equal. On the left hand side you have 3 and the right hand side you only have 1 since 1² =1.
So add 2 to the right hand side since 1²+2 would give you 3.
➣ Thus, x²-2x+3
= x²-2(x)(1)+1² +2
= (x-1)² +2
Underlined: since a²-2ab+b²= (a-b)²
And you have completed the square!
And if you were asking what to do if there's a negative sign at the coefficient of the x² term, factorise the negative sign out at the first step.
For example,
-x² +2x -3
= -(x² -2x +3)
= -[(x-1)² +2]
= -(x-1)² -2
But of course if the quadratic equation equals to zero, then divide throughout by -1.
If the coefficient of the x term is positive, then what we are looking at at step 2 is a² +2ab +b²= (a+b)²
Do feel free to ask if you still have any questions :)