Key Concepts in Completing the Square

Solving Quadratics by Completing the Square

• Transformational Form to Standard Form

  • Convert the equation:
    $y = 2(x - 3)² + 5$
  • Step 1: Expand the square:
    $y = 2(x² - 6x + 9) + 5$
  • Step 2: Distribute the coefficient 2:
    $y = 2x² - 12x + 18 + 5$
  • Step 3: Combine like terms:
    $y = 2x² - 12x + 23$

• Completing the Square

  • A quadratic in the form $x² + bx$ can be transformed into a perfect square using:

Steps to Complete the Square

1. Find half of b:

   • Identify the coefficient of x in $x² + bx$
   • Example: For $b = 4$, half is $2$.

2. Square the result:

   • Square the half found in Step 1.
   • Example: $2² = 4$.

3. Add the square to the expression:

   • Combine with the original expression:
     $x² + bx + (2)² = (x + 2)²$