Introduction

1. Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations by transforming the left side of the equation into a perfect square trinomial and using square roots to solve. 5.2.3: Completing the Square

2. Key Concepts When the binomial (x + a) is squared, the resulting perfect square trinomial is x2+ 2ax + a2. When the binomial (ax + b) is squared, the resulting perfect square trinomial is a2x2+ 2abx + b2. 5.2.3: Completing the Square

3. Key Concepts, continued 5.2.3: Completing the Square

4. Common Errors/Misconceptions neglecting to add the c term of the perfect square trinomial to both sides not isolating x after squaring both sides forgetting that, when taking the square root, both the positive and negative roots must be considered (±) 5.2.3: Completing the Square

5. Guided Practice Example 2 Solve x2+ 6x + 4 = 0 by completing the square. 5.2.3: Completing the Square

6. Guided Practice: Example 2, continued Determine if x2 + 6x + 4 is a perfect square trinomial. Take half of the value of b and then square the result. If this is equal to the value of c, then the expression is a perfect square trinomial. x2+ 6x + 4 is not a perfect square trinomial because the square of half of 6 is not 4. 5.2.3: Completing the Square

7. Guided Practice: Example 2, continued Complete the square. 5.2.3: Completing the Square

8. Guided Practice: Example 2, continued Express the perfect square trinomial as the square of a binomial. Half of b is 3, so the left side of the equation can be written as (x + 3)2. (x + 3)2= 5 5.2.3: Completing the Square

9. Guided Practice: Example 2, continued Isolate x. 5.2.3: Completing the Square

10. Guided Practice: Example 2, continued Determine the solution(s). The equation x2 + 6x + 4 = 0 has two solutions, ✔ 5.2.3: Completing the Square

11. Guided Practice: Example 2, continued 5.2.3: Completing the Square

12. Guided Practice Example 3 Solve 5x2– 50x – 120 = 0 by completing the square. 5.2.3: Completing the Square

13. Guided Practice: Example 3, continued Determine if 5x2 – 50x – 120 = 0 is a perfect square trinomial. The leading coefficient is not 1. First divide both sides of the equation by 5 so that a = 1. 5.2.3: Completing the Square

14. Guided Practice: Example 3, continued Now that the leading coefficient is 1, take half of the value of b and then square the result. If the expression is equal to the value of c, then it is a perfect square trinomial. 5x2 – 50x – 120 = 0 is not a perfect square trinomial because the square of half of –10 is not –24. 5.2.3: Completing the Square

15. Guided Practice: Example 3, continued Complete the square. 5.2.3: Completing the Square

16. Guided Practice: Example 3, continued Express the perfect square trinomial as the square of a binomial. Half of b is –5, so the left side of the equation can be written as (x – 5)2. (x – 5)2 = 49 5.2.3: Completing the Square

17. Guided Practice: Example 3, continued Isolate x. 5.2.3: Completing the Square

18. Guided Practice: Example 3, continued Determine the solution(s). The equation 5x2 – 50x – 120 = 0 has two solutions, x = –2 or x = 12. ✔ 5.2.3: Completing the Square

19. Guided Practice: Example 3, continued 5.2.3: Completing the Square