Write the expression 
9x^2-9x+1
in the form a(x+b)^2+c, where a, b and c are real numbers

       9x² - 9x + 1

It's already arranged in descending order, so we first

1. Factor only the coefficient of x² out of only the first
   two terms.  Use brackets instead of parentheses because
   there are going to be some parentheses inside the
   brackets:

       9[x² - x] + 1

2. To the side, 
   a, Multiply the coefficient of x in the brackets, which
      is -1, by 1/2

                                           -1·1/2 = -1/2

   b.  Square this result:
      
                                           (-1/2)² = +1/4

3. Add and subtract this amount inside the brackets:

      9[x² - x + 1/4 - 1/4] + 1
        
4. Factor only the first three terms.

   9[(x - 1/2)(x - 1/2) - 1/4] + 1

   Write that parentheses only once squared

   9[(x - 1/2)² - 1/4] + 1

   Now I'll do some coloring:

   9[(x - 1/2)² - 1/4] + 1

5. Remove the brackets by distributing 9 first
   into the blue thing, and then into the red
   thing, leaving the entire blue thing intact.

    9(x - 1/2)² - 9/4 + 1

6. Add the two numbers on the right end by
   getting the LCD and rewriting using the LCD:
   
   9(x - 1/2)² - 9/4 + 4/4

   9(x - 1/2)² - 5/4
    
That's it!

   a(x + b)² + c with a = 9, b = -1/2 and c = -5/4.

Edwin