1. 2.2: Solving Equations Through Various Methods
Page 87 3) 3 7)
{≈–2.4265}
11)
{≈–1.4751, ≈1.2372}
15)
{≈–1.3794, ≈1.6030}
19)
{≈–2.1149, ≈0.2541, ≈1.8608}
23)
{≈–1.7521}
27)
{0, ≈2.2074}
31)
{≈–0.6514, ≈1.1513}
33)
{≈7.0334}
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2.2: Solving Equations Through Various Methods

2. Solving Equations Through Various Methods
Section 2.2
Pre-Calculus AB Pre-AP/Dual, Revised ©2014
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2.2: Solving Equations Through Various Methods

3. Solve using Square Root Property
How do you get rid of the square root of an equation?
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2.2: Solving Equations Through Various Methods

4. 2.2: Solving Equations Through Various Methods
Square Root Property
Isolate the variable on one side and number onto other side
Get rid of the squared number by square rooting it
Solve and do not forget the PLUS-OR-MINUS symbol
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2.2: Solving Equations Through Various Methods

5. 2.2: Solving Equations Through Various Methods
Example 1
Solve using Square Root Method, 3x2 + 1 = 16
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2.2: Solving Equations Through Various Methods

6. 2.2: Solving Equations Through Various Methods
Example 2
Solve using Square Root Method, 2x = 5
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2.2: Solving Equations Through Various Methods

7. 2.2: Solving Equations Through Various Methods
Example 3
Solve using Square Root Method, (x – 3)2 = 5
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2.2: Solving Equations Through Various Methods

8. 2.2: Solving Equations Through Various Methods
Your Turn
Solve using Square Root Method, 3x2 – 4 = 68
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2.2: Solving Equations Through Various Methods

9. 2.2: Solving Equations Through Various Methods
Discriminant Formula
This is the part of the quadratic formula under the radical, b2 – 4ac (KNOW IT)
There are 3 types of answers that you can get:
2 positive roots if the discriminant equals POSITIVE NUMBER
1 real double root if the discriminant equals ZERO
2 imaginary roots if the discriminant equals NEGATIVE NUMBER
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2.2: Solving Equations Through Various Methods

10. 2.2: Solving Equations Through Various Methods
Example 4
Determine the determinant and nature of the roots, x2 – 4x + 3 = 0 A B C 1 –4 3
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2.2: Solving Equations Through Various Methods

11. 2.2: Solving Equations Through Various Methods
Example 5
Determine the determinant and nature of the roots, 9x2 + 12x + 4 = 0
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2.2: Solving Equations Through Various Methods

12. 2.2: Solving Equations Through Various Methods
Your Turn
Determine the nature of the roots, –3x2 + 7x – 9 = 0
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2.2: Solving Equations Through Various Methods

13. Example 4 Solve through factoring, 0 = x2 – 5x + 4 11/22/2018 4:01 AM
10.5: Solving Equations

14. 2.2: Solving Equations Through Various Methods
Example 5
Solve by factoring, 15b2 – 22b + 8 = 0 TS TP
–22
+120
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2.2: Solving Equations Through Various Methods

15. 2.2: Solving Equations Through Various Methods
Your Turn
Solve by factoring, 5p2 + 19p + 12 = 0
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2.2: Solving Equations Through Various Methods

16. 2.2: Solving Equations Through Various Methods
Quadratic Formula
Quadratic Formula is another way of finding x
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2.2: Solving Equations Through Various Methods

17. Steps for Quadratic Formula
Make sure the equation equal to ZERO
Identify A, B, C and plug into equation
Evaluate the square root, remember there are 2 answers.
Check your work
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2.2: Solving Equations Through Various Methods

18. 2.2: Solving Equations Through Various Methods
Quadratic Formula
Quadratic Formula is another way of finding x
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2.2: Solving Equations Through Various Methods

19. 2.2: Solving Equations Through Various Methods
Example 7
Solve x2 – 4x + 3 = 0 using Quadratic Formula A: 1, B: –4, C: 3
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2.2: Solving Equations Through Various Methods

20. 2.2: Solving Equations Through Various Methods
Example 7
Solve x2 – 4x + 3 = 0 using Quadratic Formula A: 1, B: –4, C: 3
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2.2: Solving Equations Through Various Methods

21. 2.2: Solving Equations Through Various Methods
Example 8
Solve –2x2 + 4x – 3 = 0 using Quadratic Formula
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2.2: Solving Equations Through Various Methods

22. 2.2: Solving Equations Through Various Methods
Your Turn
Solve –9x – 1 = –9x2 using Quadratic Formula
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2.2: Solving Equations Through Various Methods

23. Steps for Completing the Square
Put terms with variables on one side and CONSTANT to the other side
Make sure the side is in DESCENDING order
Standard Form, Ax2 + Bx + C = 0 where A = 1
Identify the coefficient which is raised to the first power, DIVIDE the term by 2, and SQUARE the number (it will always be positive)
ADD to both sides to the equation
Put the equation into FACTORED form (Vertex Form)
SQUARE ROOT both sides and cancel the binomial
Solve for x and check
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2.2: Solving Equations Through Various Methods

24. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 1: Put terms with variables on one side and terms without variables onto other side
Make sure the side is in DESCENDING order
Standard Form, Ax2 + Bx + C = 0 where A = 1
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2.2: Solving Equations Through Various Methods

25. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 2: Identify the coefficient which is raised to the first power and divide the term by 2
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2.2: Solving Equations Through Various Methods

26. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 3: Take the second term, divide the term by 2, and square that number
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2.2: Solving Equations Through Various Methods

27. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 4: Add to both sides to the equation
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2.2: Solving Equations Through Various Methods

28. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 5: Put the equation into factored form (Vertex Form)
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2.2: Solving Equations Through Various Methods

29. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 6: Square Root both sides and cancel the binomial
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2.2: Solving Equations Through Various Methods

30. 2.2: Solving Equations Through Various Methods
Example 9
Solve x2 – 4x + 3 = 0 using completing the square
Step 7: Solve and Check
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2.2: Solving Equations Through Various Methods

31. 2.2: Solving Equations Through Various Methods
Example 10
Solve x2 = 12x – 20 using completing the square
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2.2: Solving Equations Through Various Methods

32. 2.2: Solving Equations Through Various Methods
Example 11
Solve 2x2 + 4x = –9 using completing the square
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2.2: Solving Equations Through Various Methods

33. 2.2: Solving Equations Through Various Methods
Your Turn
Solve x2 – 4x = –2 using completing the square
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2.2: Solving Equations Through Various Methods

34. 2.2: Solving Equations Through Various Methods
Assignment
Page odd, EOO For answers involving imaginary solutions, provide the complex number and not “no real solutions”
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2.2: Solving Equations Through Various Methods