1. Completing the Square with Parabolas

2. Perfect Square Trinomials
Examples
x2 + 6x + 9
x2 - 10x + 25
x2 + 12x + 36

3. Creating a Perfect Square Trinomial
In the following perfect square trinomial, the constant term is missing X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term.
(14/2) X2 + 14x + 49

4. Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___
x2 - 4x + ___
x2 + 5x + ___
100 4
25/4

5. Completing the Square with Parabolas
Example 1: Put the following parabola in vertex form by
completing the square:
Step 1: Move the constant, c, to the other side.
𝑦= 𝑥 2 +8𝑥−20
𝑦+20= 𝑥 2 +8𝑥

6. Completing the Square with Parabolas
Step 2: Find 𝑏 2 .
𝑦+20= 𝑥 2 +8𝑥
𝑏 2 = 8 2 =4

7. Completing the Square with Parabolas
Step 3: Take 𝑏 2 2
𝑏 2 = 8 2 =4
4 2 =16

8. Completing the Square with Parabolas
Step 4:
Add 𝑏 to both sides of the equation.
𝑦+20= 𝑥 2 +8𝑥
+16
+16

9. Completing the Square with Parabolas
𝑦+36=𝑥 2 +8𝑥+16
𝑦+36= 𝑥+4 2
Step 5: Factor the perfect square trinomial.

10. Completing the Square with Parabolas
𝑦+36= 𝑥+4 2
𝑦= 𝑥+4 2 −36
Step 6:
Move the constant back to the right side.
−36
−36
Can you identify the vertex of this parabola?

11. Completing the Square with Parabolas
Example 2: Put the following parabola in vertex form by
completing the square:
Step 1: Move the constant, c, to the other side.
𝑦= 𝑥 2 −4𝑥+1
− −1
𝑦−1= 𝑥 2 −4𝑥

12. Completing the Square with Parabolas
Step 2: Find 𝑏 2 .
𝑦−1= 𝑥 2 −4𝑥
𝑏 2 = −4 2 =−2

13. Completing the Square with Parabolas
Step 3: Take 𝑏 2 2
𝑏 2 = −4 2 =−2
(−2) 2 =4

14. Completing the Square with Parabolas
Step 4:
Add 𝑏 to both sides of the equation.
𝑦−1= 𝑥 2 −4𝑥 +4 +4

15. Completing the Square with Parabolas
𝑦+3=𝑥 2 −4𝑥+4
𝑦+3= 𝑥−2 2
Step 5: Factor the perfect square trinomial.

16. Completing the Square with Parabolas
𝑦+3= 𝑥−2 2
𝑦= 𝑥−2 2 −3
Step 6:
Move the constant back to the right side. −3 −3
Can you identify the vertex of this parabola?

17. Completing the Square with Parabolas
Example 3: Put the following parabola in vertex form by
completing the square:
Step 1: Move the constant, c, to the other side.
𝑥= 𝑦 2 +12𝑦+25
− −25
𝑥−25= 𝑦 2 +12𝑦

18. Completing the Square with Parabolas
Step 2: Find 𝑏 2 .
𝑥−25= 𝑦 2 +12𝑦
𝑏 2 = 12 2 =6

19. Completing the Square with Parabolas
Step 3: Take 𝑏 2 2
𝑏 2 = 12 2 =6
6 2 =36

20. Completing the Square with Parabolas
Step 4:
Add 𝑏 to both sides of the equation.
𝑥−25= 𝑦 2 +12𝑦
+36
+36

21. Completing the Square with Parabolas
𝑥+11=𝑦 2 +12𝑦+36
𝑥+11= 𝑦+6 2
Step 5: Factor the perfect square trinomial.

22. Completing the Square with Parabolas
𝑥+11= 𝑦+6 2
𝑥= 𝑦+4 2 −11
Step 6:
Move the constant back to the right side.
−11
−11
Can you identify the vertex of this parabola?

23. Completing the Square with Parabolas
Example 4: Put the following parabola in vertex form by
completing the square:
Step 1: Move the constant, c, to the other side.
𝑦=4 𝑥 2 −24𝑥−5 +5 +5
𝑦+5= 4𝑥 2 −24𝑥

24. Completing the Square with Parabolas
Step 2: Find 𝑏 2 .
**In order to complete the square, 𝑎=1.
**Before we can find 𝑏 2 , we must factor out the 𝑎=4.
𝑦+5= 4𝑥 2 −24𝑥 𝑦+5=4( 𝑥 2 −6𝑥)
𝑏 2 = −6 2 =−3

25. Completing the Square with Parabolas
Step 3: Take 𝑏 2 2
𝑏 2 = −6 2 =−3
(−3) 2 =9

26. Completing the Square with Parabolas
Step 4:
Add 𝑏 to both sides of the equation.
𝑦+5= 4(𝑥 2 −6𝑥
+36
+9)

27. Completing the Square with Parabolas
𝑦+41=4(𝑥 2 −6𝑥+9)
𝑦+41= 4 𝑥−3 2
Step 5: Factor the perfect square trinomial.

28. Completing the Square with Parabolas
𝑦+41= 4 𝑥−3 2
𝑦= 4 𝑥−3 2 −41
Step 6:
Move the constant back to the right side.
−41
−41
Can you identify the vertex of this parabola?

29. Completing the Square with Parabolas
𝑥=2 𝑦 2 −28𝑦+10
You Try!
Put the following parabola in vertex form by
completing the square:
Can you identify the vertex of this parabola?