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