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

Imaginary Numbers and Complex Numbers 𝑖= −1 𝑖 2 =−1 Imaginary numbers allow us to simplify radicals with no real solutions. −4 = −1 4 =2𝑖 −3 = −1 3 =𝑖 3 −20 = −1 4 5 =2𝑖 5 A complex number written in standard form is a number 𝑎+𝑏𝑖 where 𝑎 and 𝑏 are real numbers. The number 𝑎 is the real part, and the number 𝑏𝑖 is the imaginary part. Real Numbers: 5 −1 𝜋 2 3 Complex Numbers: 2+3𝑖 9−5𝑖 Pure Imaginary Numbers: 3𝑖 −5𝑖

Examples 1 and 2 Example 1: Find the square root of each number a) −25 b) −72 c) −5 −9 = −1 25 = −1 36 2 =−5 −1 9 =5𝑖 =6𝑖 2 =−5∙3𝑖 =−15𝑖 Example 2: Find the values of x and y that satisfy the equation 2𝑥−7𝑖=10+𝑦𝑖 ** Set the real parts equal to each other, set the imaginary parts equal to each other. Solve the resulting equations. ** 2𝑥=10 −7𝑖=𝑦𝑖 𝑥=5 −7=𝑦 y=−7

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: 8−𝑖 +(5+4𝑖) 7−6𝑖 −(3−6𝑖) 13− 2+7𝑖 +5𝑖 = 8+5 +(−𝑖+4𝑖) =13+3𝑖 = 7−3 +(−6𝑖−(−6𝑖)) =4+0𝑖 =4 =13−2−7𝑖+5𝑖 =11−2𝑖

Multiplying Complex Numbers Example 5: Multiply. Write the answer in standard form. a) 4𝑖(−6+𝑖) b) (9−2𝑖)(−4+7𝑖) ** 𝑖∙𝑖= 𝑖 2 =−1 ** =−36+63𝑖+8𝑖−14 𝑖 2 =−24𝑖+4 𝑖 2 =−36+71𝑖−14(−1) =−24𝑖+4(−1) =−36+71𝑖+14 =−24𝑖−4 =−22+71𝑖 =−4−24𝑖

Solving Quadratic Equations Example 6: Solve a) 𝑥 2 +4=0 b) 2 𝑥 2 −11=−47 𝑥 2 =−4 2 𝑥 2 =−36 𝑥= −4 𝑥 2 =−18 𝑥= −1 4 𝑥= −18 𝑥= −1 9 2 𝑥=±2𝑖 𝑥=±3𝑖 2

Finding Zeros of Quadratic Functions Example 7: Find the zeros of 𝑓 𝑥 =4 𝑥 2 +20 0=4 𝑥 2 +20 −20=4 𝑥 2 −5= 𝑥 2 𝑥= −5 𝑥= −1 5 𝑥=±𝑖 5 The zeros of the function are 𝑖 5 and −𝑖 5

Practice Find the square root of the number. −4 2) −12 3) − −36 4) 2 −54 Find the values of x and y that satisfy the equation. 5) 𝑥+3𝑖=9−𝑖𝑦 6) 9+4𝑦𝑖=−2𝑥+3𝑖 Perform the operation. Write the answer in standard form. 8) 9−𝑖 +(−6+7𝑖) 9) 3+7𝑖 −(8−2𝑖) 10) −4− 1+𝑖 −(5+9𝑖) 11) (−3𝑖)(10𝑖) 12) 𝑖(8−𝑖) 13) (3+𝑖)(5−𝑖) Answers: 1) 2𝑖 2) 2𝑖 6 3) −6𝑖 4) 6𝑖 6 5) 𝑥=9, 𝑦=−3 6) 𝑥=−4.5, 𝑦=0.75 8) 3+6𝑖 9) −5+9𝑖 10) −10−10𝑖 11) 30 12) 1+8𝑖 13) 16+2𝑖