1. Simplify – Do not use a calculator
1) √24
2) √80
Hint: Start by finding the Prime Factorization of each number

2. 4.6 Solving Quadratic Equations by Completing the Square Learning Target: I can solve equations by completing the square

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

4. Perfect
Square On
One side
Review
Solve for x
Take
Square Root of
BOTH SIDES
When you take the square root, You MUST consider the Positive and Negative answers.

5. But what happens if you DON’T have a perfect square on one side…….
Solve for x
Take
Square Root of
BOTH SIDES
But what happens if you DON’T have a perfect square on one side…….
You make it a Perfect Square
Use the relations on next slide…

6. To expand a perfect square binomial:
Short Cut
To expand a perfect square binomial:
We can use this relationship to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.
Then

7. Make this a perfect square trinomial
Take ½ middle term
Then square it
The resulting trinomial is called a perfect square trinomial,
which can be factored into a perfect square binomial.

8. Solve by completing the square
Make one side a perfect square
Add a blank to both sides
Divide “b” by 2
4. Square that answer.
Add it to both sides
Factor 1st side
Square root both sides
Solve for x
Solve by completing the square 1.

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

10. Steps to solve by completing the square
1.) If the quadratic does not factor, move the constant to the other side of the equation Ex: x²-4x -7 =0 x²-4x=7 2.) Work with the x²+ x side of the equation and complete the square by taking ½ of the coefficient of x and squaring Ex. x² -4x 4/2= 2²=4 3.) Add the number you got to complete the square to both sides of the equation Ex: x² -4x +4 = )Simplify your trinomial square Ex: (x-2)² =11 5.)Take the square root of both sides of the equation Ex: x-2 =±√11 6.) Solve for x Ex: x=2±√11

11. Solving Quadratic Equations by Completing the Square
Solve the following equation by completing the square:
Step 1: Set quadratic equation equal to zero

12. Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square. Add that term that is equal to zero into the equation.
X2 + 8x + ____ + _____ -20 = 0 16
-16

13. Solving Quadratic Equations by Completing the Square
Step 3: Factor the terms that create the perfect square trinomial. Simplify the other 2 terms of the equation.
X2 + 8x + ____ + _____ -20 = 0 16
-16
(x + 4)(x + 4) - 36 = 0
(x + 4) = 0
Note: This is vertex form of the equation
y =(x + 4)2 - 36

14. Solving Quadratic Equations by Completing the Square
Step 4: Move the constant term and isolate the square binomial.
(x + 4) = 0
(x + 4)2 = 36

15. Solving Quadratic Equations by Completing the Square
Step 5: Take the square root of each side

16. Solving Quadratic Equations by Completing the Square
Step 6: Set up the two possibilities and solve
Was there an easier way?

17. Solve by Completing the Square+9

18. Solve by Completing the Square
+121

19. Solve by Completing the Square+1

20. Solve by Completing the Square
+25

21. Solve by Completing the Square
+16

22. Solve by Completing the Square+9

23. Assignment pg
Homework– p odds

24. Completing the Square-Example #2
Solve the following equation by completing the square:
Step 1: If the lead coefficient is not 1, factor the lead coefficient from the a and b terms.
2x2 + 12x - 5 = 0
2(x2 + 6x) - 5 = 0

25. Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square.
(remember add in zero)
The quadratic coefficient must be equal to 1 before you complete the square, so you must divide the first 2 terms by the quadratic coefficient first.
2(x2 + 6x + ___) +___ - 5 = 0 9

26. Solving Quadratic Equations by Completing the Square
Step 2: What makes out of the parenthesis zero?
2(x2 + 6x + ___) +___ - 5 = 0 9
-18

27. Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial in the equation. Combine the other two terms.
2(x2 + 6x + ___) +___ - 5 = 0
2(x + 3) = 0 9
-18

28. Solving Quadratic Equations by Completing the Square
Step 4: Move the constant to the right side of the equation and solve.
Isolate the perfect square
Take the square root of both sides

29. Solving Quadratic Equations by Completing the Square
Step 4 continued:

30. Solving Quadratic Equations by Completing the Square
Try the following examples. Do your work on your paper and then check your answers.
x2 + 2x - 63 = 0
x2 - 10x - 15 = 0
2x2 - 6x - 1 = 0