Complex Numbers and Quadratic Equations
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- Alberta Ward
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1 Chapter 16: Quadratic Equations 99 Chapter 16 Complex Numbers and Quadratic Equations Review: Simplifying Radicals The Product Rule for Radicals For any non negative radicands Eand F,» EÜ» Fú» EÜF Simplify by factoring and applying the Product Rule for Radicals. Hint: When simplifying square roots, look for perfect squares!fl"fl%fl*fl"'fl&fl$'fl%*fl'%fl)"fl"!!fl""fl"%%fl"'*fl"*'fl&fl 21.»") 22.»" 23.»! 24.»% 25.»( 26.»%! 27.»&! 28.»') Basic Concepts of Complex Numbers The Imaginary Unit 3 Complex Number 3ú» Å", and therefore, 3 úå" +Ä,3 Write as the product of a real number and » Å"' 30.» Å%
2 100 Mr. Getso s Algebra Notes Operations on Complex Numbers Multply or divide, as indicated. Simplify each answer. 31.» Å( Ü» Å( 32.» Å' Ü» Å"! 33.» Å! 34.» Å%)» Å»% Write in standard form +Ä,3. Å)Ä» Å%) "&Å» Å(& % &
3 Chapter 16: Quadratic Equations 101 Find the sum. 37. aä&3bä a$ä(3b 38. aå'ä$3bäaåå&3b Find the difference. 39. aå%ä$3bå a'å(3b 40. aå"!ä(3bå a&å$3b Find the product. 41. aå$3 ba$ä%3 b 42. a%ä$3 b
4 102 Mr. Getso s Algebra Notes Properties of Complex Conjugates For real numbers + and,, a+ä,3 ba+å,3b ú+ Ä, 43. a'ä&3 ba'å&3 b 44. a&å3 ba&ä3b Write the quotient in standard form +Ä,3. ( ) ( ) 45. $Ä(3 ú $Ä(3 Ü 46. $Ä3 'Ä&3 'Ä&3 &Å3 ú
5 Chapter 16: Quadratic Equations Solving Quadratic Equations by the Square Root Property Review: Simplifying Radicals The Product Rule for Radicals For any non negative radicands Eand F,» EÜ» Fú» EÜF Simplify by factoring by applying the Product Rule for Radicals. Hint: When simplifying, look for perfect squares!fl"fl%fl*fl"'fl&fl$'fl%*fl'%fl)"fl"!!fl""fl"%%fl"'*fl"*'fl&fl 1.») 2.»" 3.»! 4.»% 5.»( 6.»%! 7.»&! 8.»'! 9.»)! 10.»*!
6 104 Mr. Getso s Algebra Notes Using the Square Root Property: B ú- Square Root Property If B ú+ for +!, then Bú» + or BúÅ» +. Use the square root property to solve each of the following: 11. B ú* 12. B Å"' ú! 13. B ú" 14. B Å& ú! 15. B ú"!! 16. B Å'% ú! 17. B ú) 18. B Å" ú!
7 Chapter 16: Quadratic Equations 105 Using the Square Root Property: +B ú- 19. % B ú* 20. * B Å& ú! 21. "' B ú" 22. & B ú% 23. %* B ú$ 24. $' B ú& 25. B ú) 26. $' B ú"
8 106 Mr. Getso s Algebra Notes Using the Square Root Property: abä+ b ú- 27. abå$ b ú"' 28. abå% b Å%* ú! 29. abä" b ú* 30. abå& b Å)" ú! 31. abä( b ú) 32. abå"& b Å") ú!
9 Chapter 16: Quadratic Equations 107 Review Basic Concepts of Complex Numbers The Imaginary Unit 3 Complex Number 3ú» Å", and therefore, 3 úå" +Ä,3 Write as the product of a real number and » Å'% 34.» Å'! ú Operations on Complex Numbers Write in standard form +Ä,3. Å)Ä» Å") Å)» Å") "&Å» Å(& 35. ú Ä 36. ú % % % & ú Å Ä ú Å Ä ú Å Ä 3»") % 3»'%» % )3» % ú Å Ä3» Using the Square Root Property: abä+ b úå- 37. abå" b úå 38. abä$ b Ä&ú!
10 108 Mr. Getso s Algebra Notes Using the Square Root Property: a+bä, b ú- 39. a& BÅ b ú"! 40. a% BÄ" b ú"& 41. a$ BÅ% b Ä&ú! 42. a& BÄ% b ú$
11 Chapter 16: Quadratic Equations Solving Quadratic Equations by Completing the Square Review: Using the Square Root Property Square Root Property If B ú+ for +!, then Bú» + or BúÅ» +. Use the square root property to solve. 1. abä$ b ú$ 2. abå% b ú( 3. abä" b ú) 4. abå& b ú") 5. a& BÅ b ú! 6. a% BÄ( b ú) 7. abå" b úå 8. abä$ b úå&
12 110 Mr. Getso s Algebra Notes Review: Perfect Square Trinomials Multiply each of the following: 9. abä" b úb ÄBÄ" 10. abå b ú 11. abå$ b úb Å'BÄ* 12. abå% b ú 13. abä& b úb Ä"!BÄ& 14. abä' b ú 15. abå( b úb Å"%BÄ%* 16. abå) b ú 17. abä* b úb Ä")BÄ)" 18. abä"! b ú What number must be added to eachexpression to obtain a perfect square trinomial? 19. B Å"%BÄ 20. B Å!BÄ 21. B Å"'BÄ 22. B Ä%BÄ 23. B Å")BÄ 24. B Ä%!BÄ 25. B ÄBÄ 26. B Å&!BÄ 27. B ÄBÄ 28. B Ä$BÄ " 29. B Ä BÄ 30. B Ä BÄ $
13 Chapter 16: Quadratic Equations 111 Completing the Square to Solve B Ä,BÄ-ú! Completing the Square, To complete the square on B Ä,B, add å ç. Solve by completing the square. 31. B Å"!BÄ"% ú! 32. B Ä)BÄ"ú! Bú &Ñ»"" 33. B Å"!B úå"% 34. B Å"%B úå$
14 112 Mr. Getso s Algebra Notes 16.3 Solving Quadratic Equations by the Quadratic Formula Review: Using the Square Root Property Square Root Property If B ú+ for +!, then Bú» + or BúÅ» +. Use the square root property to solve. 1. abå' b ú& 2. abå"' b ú$' 3. abä$ b ú" 4. abå& b ú! 5. a& BÅ b ú%! 6. a% BÄ" b ú&!
15 Chapter 16: Quadratic Equations 113 Review: Completing the Square to Solve B Ä,BÄ-ú! Completing the Square, To complete the square on B Ä,B, add å ç. Solve by completing the square. 7. +B Ä,BÄ-ú! Standard Form: +B Ä,BÄ-ú!fl +! Write in Standard Form and find +,,, and -fi 8. B Å)BÅ&ú! 9. B Å$BÄú! +ú,ú -ú +ú,ú -ú 10. B Å&B ú! 11. B Ä(B ú%ä(b +ú,ú -ú +ú,ú -ú
16 114 Mr. Getso s Algebra Notes Using the Quadratic Formula Quadratic Formula If +,,, and - are real numbers and +!, a quadratic equation written in the form +B Ä,BÄ-ú! has solutions Bú Å,Ñ», Å%+-. + Solve by using the quadratic formula. 12. B Ä'BÄ%ú! 13. B Ä)BÄ%ú! 14. B ÄB ú$& 15. B ú")bå)"
17 Chapter 16: Quadratic Equations Graphing Quadratic Equations in Two Variables 1. Graph: CúB ÄBÅ$ y Step 0 +ú,,ú, -ú 6 C-intercept Step 1 If + is positive,. If + is negative, x Step 2 Find B-intercept(s): Set Cú! and solve Step 3 Identify the C-intercept. Step 4 Find the middle : Bú Å, + Å, Step 5 Find the vertex: Use Bú and find C. +
18 116 Mr. Getso s Algebra Notes 2. Graph: CúÅB ÅBÄ$ y Step 0 +ú,,ú, -ú 6 C-intercept Step 1 If + is positive,. If + is negative, x Step 2 Find B-intercept(s): Set Cú! and solve Step 3 Identify the C-intercept. Step 4 Find the middle : Bú Å, + Å, Step 5 Find the vertex: Use Bú and find C. +
19 Chapter 16: Quadratic Equations Graph: CúB Ä&BÄ% y Step 0 +ú,,ú, -ú 6 C-intercept Step 1 If + is positive,. If + is negative, x Step 2 Find B-intercept(s): Set Cú! and solve Step 3 Identify the C-intercept. Step 4 Find the middle : Bú Å, + Å, Step 5 Find the vertex: Use Bú and find C. +
20 118 Mr. Getso s Algebra Notes 4. Graph: CúB ÅBÅ' y Step 0 +ú,,ú, -ú 6 C-intercept Step 1 If + is positive,. If + is negative, x Step 2 Find B-intercept(s): Set Cú! and solve Step 3 Identify the C-intercept. Step 4 Find the middle : Bú Å, + Å, Step 5 Find the vertex: Use Bú and find C. +
Chapter 11: Factoring Polynomials Greatest Common Factor. ca d Factoring When Terms Have a Common Factor. Factor completely. B Å'B * # "& B Ä'B
Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes Spring 2015 11.1 Greatest Common Factor ca d Factoring When Terms Have a Common Factor 1. Factor completely. B Å'B * "& 2. B Ä B $ ( 3. B ÅB
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