Lecture Notes: Solving Quadratic Equations by Completing the Square

Introduction

• Presenter: Fiori
• Topic: Solving quadratic equations by completing the square.

Understanding Completing the Square

• Binomial Definition: If (x^2 + bx) is a binomial:
  • Add ((\frac{b}{2})^2) to form a perfect square trinomial: [x^2 + bx + (\frac{b}{2})^2 = (x + \frac{b}{2})^2]
• Application: Solve any quadratic equation using this method with the square root property.

Steps to Solve Quadratic Equations

1. Standard Form: (ax^2 + bx + c = 0)
2. Step 1: If (a \neq 1), divide all terms by (a) to make the coefficient of (x^2) equal to 1.
3. Step 2: Isolate variable terms on one side and constants on the other.
4. Step 3: Complete the square:
   • Part A: Add the square of half the coefficient of (x) to both sides.
   • Part B: Factor the resulting trinomial.
5. Step 4: Use the square root property and solve for (x).

Example Problem

• Equation: (2x^2 + 3x - 4 = 0)
  • Step 1: Divide by 2: (x^2 + \frac{3}{2}x - 2 = 0)
  • Step 2: Add 2 to both sides: (x^2 + \frac{3}{2}x = 2)
  • Step 3: Complete the square:
    • Half of (\frac{3}{2}) is (\frac{3}{4}).
    • Add ((\frac{3}{4})^2): (x^2 + \frac{3}{2}x + (\frac{3}{4})^2 = 2 + (\frac{3}{4})^2)
  • Step 4: Final equation: [(x + \frac{3}{4})^2 = \frac{41}{16}]
  • Solution: Solve for (x) using the square root property: [x = -\frac{3}{4} \pm \frac{\sqrt{41}}{4}]

Practice Problem

• Equation: (x^2 + 4x - 1 = 0)
  • Steps:
    1. Add 1 to both sides: (x^2 + 4x = 1)
    2. Complete the square:
       • Half of 4 is 2, square it to get 4.
       • New equation: (x^2 + 4x + 4 = 1 + 4)
       • Factor to: ((x + 2)^2 = 5)
    3. Apply square root property: (x + 2 = \pm \sqrt{5})
    4. Solve for (x):
       • (x = -2 \pm \sqrt{5})

Conclusion

• Key Technique: Remember to add the square of half the coefficient of (x) to both sides to form a perfect square trinomial.
• Problem Solving Tip: Use inverse operations to move terms across the equal sign during the solving process.