Lesson 3.3 - Completing the Square

Completing the Square

  • Learning Objective: Solve quadratic equations and rewrite quadratic functions using completing the square.
  • Success Criteria:
    • Solve quadratic equations using square roots.
    • Solve via completing the square.
    • Write quadratic functions in vertex form.

Methods of Completing the Square

  • A perfect square trinomial can be expressed as x^2 + bx + c = (x + d)^2.
  • Determine value of c required for a perfect square trinomial via:
    • c = \left(\frac{b}{2}\right)^2.

Key Steps in Solving Quadratic Equations

  1. Write the equation in the form x^2 + bx = d.
  2. Add \left(\frac{b}{2}\right)^2 to both sides of the equation.
  3. Rewrite the left side as a square of a binomial.
  4. Take the square root of both sides and solve for x.

Writing Quadratic Functions in Vertex Form

  • Vertex form is given by y = a(x - h)^2 + k.
  • Find vertex through completing the square:
    • Transform y = x^2 + bx + c into vertex form by adjusting constants.

Important Examples

  • Example 1: Solve x^2 - 16x + 64 = 100 using square roots.
  • Example 2: Complete the square for x^2 + 14x:
    • Find \left(\frac{14}{2}\right)^2 = 49.

Quadratic Equation Solutions

  • For equations where a=1, apply completing the square directly.
  • For equations where a \neq 1, first divide by a before completing the square.

Additional Notes

  • The solutions can involve complex numbers if the square root of a negative is taken.
  • Use graphical methods to check the validity of solutions.