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Section 3.2 Beginning on page 104

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1 Section 3.2 Beginning on page 104
Complex Numbers Section 3.2 Beginning on page 104

2 Imaginary Numbers and Complex Numbers
𝑖= βˆ’ 𝑖 2 =βˆ’1 Imaginary numbers allow us to simplify radicals with no real solutions. βˆ’4 = βˆ’1 4 =2𝑖 βˆ’3 = βˆ’1 3 =𝑖 3 βˆ’20 = βˆ’ =2𝑖 5 A complex number written in standard form is a number π‘Ž+𝑏𝑖 where π‘Ž and 𝑏 are real numbers. The number π‘Ž is the real part, and the number 𝑏𝑖 is the imaginary part. Real Numbers: βˆ’1 πœ‹ Complex Numbers: 2+3𝑖 βˆ’5𝑖 Pure Imaginary Numbers: 3𝑖 βˆ’5𝑖

3 Examples 1 and 2 Example 1: Find the square root of each number
a) βˆ’25 b) βˆ’72 c) βˆ’5 βˆ’9 = βˆ’1 25 = βˆ’ =βˆ’5 βˆ’1 9 =5𝑖 =6𝑖 2 =βˆ’5βˆ™3𝑖 =βˆ’15𝑖 Example 2: Find the values of x and y that satisfy the equation 2π‘₯βˆ’7𝑖=10+𝑦𝑖 ** Set the real parts equal to each other, set the imaginary parts equal to each other. Solve the resulting equations. ** 2π‘₯=10 βˆ’7𝑖=𝑦𝑖 π‘₯=5 βˆ’7=𝑦 y=βˆ’7

4 The Sums and Differences of Complex Numbers
To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Example 3: 8βˆ’π‘– +(5+4𝑖) 7βˆ’6𝑖 βˆ’(3βˆ’6𝑖) 13βˆ’ 2+7𝑖 +5𝑖 = 8+5 +(βˆ’π‘–+4𝑖) =13+3𝑖 = 7βˆ’3 +(βˆ’6π‘–βˆ’(βˆ’6𝑖)) =4+0𝑖 =4 =13βˆ’2βˆ’7𝑖+5𝑖 =11βˆ’2𝑖

5 Multiplying Complex Numbers
Example 5: Multiply. Write the answer in standard form. a) 4𝑖(βˆ’6+𝑖) b) (9βˆ’2𝑖)(βˆ’4+7𝑖) ** π‘–βˆ™π‘–= 𝑖 2 =βˆ’1 ** =βˆ’36+63𝑖+8π‘–βˆ’14 𝑖 2 =βˆ’24𝑖+4 𝑖 2 =βˆ’36+71π‘–βˆ’14(βˆ’1) =βˆ’24𝑖+4(βˆ’1) =βˆ’36+71𝑖+14 =βˆ’24π‘–βˆ’4 =βˆ’22+71𝑖 =βˆ’4βˆ’24𝑖

6 Solving Quadratic Equations
Example 6: Solve a) π‘₯ 2 +4=0 b) 2 π‘₯ 2 βˆ’11=βˆ’47 π‘₯ 2 =βˆ’4 2 π‘₯ 2 =βˆ’36 π‘₯= βˆ’4 π‘₯ 2 =βˆ’18 π‘₯= βˆ’1 4 π‘₯= βˆ’18 π‘₯= βˆ’ π‘₯=Β±2𝑖 π‘₯=Β±3𝑖 2

7 Finding Zeros of Quadratic Functions
Example 7: Find the zeros of 𝑓 π‘₯ =4 π‘₯ 2 +20 0=4 π‘₯ 2 +20 βˆ’20=4 π‘₯ 2 βˆ’5= π‘₯ 2 π‘₯= βˆ’5 π‘₯= βˆ’1 5 π‘₯=±𝑖 5 The zeros of the function are 𝑖 5 and βˆ’π‘– 5

8 Practice Find the square root of the number.
βˆ’4 2) βˆ’12 3) βˆ’ βˆ’36 4) 2 βˆ’54 Find the values of x and y that satisfy the equation. 5) π‘₯+3𝑖=9βˆ’π‘–π‘¦ 6) 9+4𝑦𝑖=βˆ’2π‘₯+3𝑖 Perform the operation. Write the answer in standard form. 8) 9βˆ’π‘– +(βˆ’6+7𝑖) 9) 3+7𝑖 βˆ’(8βˆ’2𝑖) 10) βˆ’4βˆ’ 1+𝑖 βˆ’(5+9𝑖) 11) (βˆ’3𝑖)(10𝑖) 12) 𝑖(8βˆ’π‘–) 13) (3+𝑖)(5βˆ’π‘–) Answers: ) 2𝑖 2) 2𝑖 ) βˆ’6𝑖 ) 6𝑖 ) π‘₯=9, 𝑦=βˆ’3 6) π‘₯=βˆ’4.5, 𝑦= ) 3+6𝑖 9) βˆ’5+9𝑖 ) βˆ’10βˆ’10𝑖 11) ) 1+8𝑖 ) 16+2𝑖


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