Question 118862
Follow these steps to convert a parabola from 
{{{f(x) = ax^2 + bx + c}}}
    to {{{f(x)= a(x-h)^2 + k}}}    

 
1. Factor out an {{{a}}} from the first two terms, and leave a space inside.
 
  {{{f(x) = a (x^2 + (b/a)x + _) + c}}}   
2. Multiply the coefficient of the {{{x}}} term by {{{1/2}}}  

   {{{(1/2)(b/a) = b/2a}}}
   
3. Square this term and add it to the space we made.

{{{f(x) = a (x^2 + (b/a)x + (b/2a)^2) + c}}}   

 4. Now, we have upset the balance of the equation. We {{{added }}}

{{{(b/2a)^2)}}} so we have {{{a(b/2a)^2)}}}  which we need to {{{subtract}}} in order to keep the balance of the equation

{{{f(x) = a (x^2 + (b/a)x + (b/2a)^2) + c - a(b/2a)^2)}}}   

5. Now, we can write the part that is in parentheses {{{(x^2 + (b/a)x + (b/2a)^2)}}}  as 

	{{{ (x+  b/2a)^2 }}}

 This is why we did all of the above steps.
   
	{{{f(x) = a (x+  b/2a)^2 + c - a(b/2a)^2)}}}   

   6. Rewrite the terms and simplify a bit. The reason for this will become apparent later on.
 {{{f(x)= a(x-(-b/2a))+ (4ac-b^2)/4a}}}    

  
   7. Now, let’s simply things by calling 

{{{h=-(b/2a)}}} and {{{k = (4ac-b^2)/4a}}}
    {{{f(x)= a(x-h)^2 + k}}}