Goals of Completing the Square

• Objective: Converting a quadratic equation into a perfect square trinomial.

• Applications: Essential for setting up the general equations of conic sections (circle, ellipse, parabola, hyperbola).

Steps for Completing the Square

1. Set the equation to zero:

   • Example: Rearranging the equation as necessary.

2. Divide by the Greatest Common Factor (GCF):

   • Simplifies the quadratic equation when applicable.

3. Move the constant to the right side:

   • Rearranging the equation facilitates breaking down the quadratic.

4. Factor the quadratic term (if leading coefficient is not equal to 1):

   • Identify the leading coefficient and adjust the equation as needed.

   • Find half of the linear coefficient, square it, and add it to both sides of the equation.

     • Example:

       • For the equation x^2 + 8x + 10 = 0, the linear coefficient is 8. Half of 8 is 4, and squaring it gives 16. This means we will adjust the equation accordingly.

5. Convert the left-hand side into a binomial squared:

   • Example:

     • From x^2 + 8x + 16 = 6, it can be rewritten as (x + 4)^2 = 6.

Examples of Completing the Square

• Single Variable Quadratics:

  • x² + 8x + 10 = 0

  • x² - 4x - 3 = 0

  • 2x² - 4x + 5 = 0

  • -3x² - 12x - 17 = 0

• Multiple Variables: Completing the square also applies when dealing with terms involving both x and y:

  • Example: x² + y² + 2x - 8y + 8 = 0

  • Example: 4x² + y² + 16x - 6y - 39 = 0

  • Example: 3x² - y² + 4x + 8y - 11 = 0

• When working with two variables, remember to group the x and y terms when applying the method.

Important Notes

• Ensure all steps are followed methodically to correctly manipulate and simplify the equations.

• Practice with various examples to master completing the square, especially with different variable combinations and coefficients.