International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each number by using the imaginary number i. = −𝟏 ∙ 𝟒𝟗 = −𝟏 ∙ 𝟏𝟒𝟒 =𝒊∙ 𝟒𝟗 =𝒊∙ 𝟏𝟒𝟒 =𝟕𝒊 =𝟏𝟐𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. −𝟐+𝟓+𝟑𝒊−𝟐𝒊 𝟑+𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. 𝟒−𝟐𝒊+𝟏−𝟑𝒊 𝟒+𝟏−𝟐𝒊−𝟑𝒊 𝟓−𝟓𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. −𝟓+𝟑𝒊+𝟖−𝟐𝒊 −𝟓+𝟖+𝟑𝒊−𝟐𝒊 𝟑+𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. −𝟐𝟎+𝟏𝟔𝒊 + 𝟏𝟓𝒊−𝟏𝟐𝒊² −𝟐𝟎+𝟑𝟏𝒊−𝟏𝟐(−𝟏) −𝟐𝟎+𝟑𝟏𝒊+𝟏𝟐 −𝟐𝟎+𝟏𝟐+𝟑𝟏𝒊 −𝟖+𝟑𝟏𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. 𝟗−𝟏𝟓𝒊 + 𝟏𝟓𝒊−𝟐𝟓𝒊² 𝟗−𝟐𝟓(−𝟏) 𝟗+𝟐𝟓 𝟑𝟒

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. 𝟔𝒊+𝟔𝒊² 𝟔𝒊+𝟔(−𝟏) 𝟔𝒊−𝟔 The i always comes last. −𝟔+𝟔𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Simplify each expression. =(𝟒 – 𝒊)(𝟒 – 𝒊) 𝟏𝟔−𝟒𝒊 − 𝟒𝒊+𝒊² 𝟏𝟔 −𝟖𝒊+(−𝟏) 𝟏𝟔 −𝟖𝒊−𝟏 𝟏𝟔−𝟏−𝟖𝒊 𝟏𝟓−𝟖𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers 𝟓+𝟕𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Write each quotient as a complex number. 11. 5 + 2𝑖 4𝑖 ∙ −𝟒𝒊 −𝟒𝒊 = −𝟐𝟎𝒊−𝟖𝒊² −𝟏𝟔𝒊² = −𝟐𝟎𝒊−𝟖(−𝟏) −𝟏𝟔(−𝟏) = −𝟐𝟎𝒊+𝟖 𝟏𝟔 = 𝟖 𝟏𝟔 + −𝟐𝟎𝒊 𝟏𝟔 = 𝟏 𝟐 − 𝟓 𝟒 𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Write each quotient as a complex number. 12. 3 − 2𝑖 4 − 3𝑖 ∙ 𝟒+𝟑𝒊 𝟒+𝟑𝒊 = (𝟑−𝟐𝒊)(𝟒+𝟑𝒊) (𝟒−𝟑𝒊)(𝟒+𝟑𝒊) = 𝟏𝟐+𝟗𝒊−𝟖𝒊−𝟔𝒊² 𝟏𝟔+𝟏𝟐𝒊−𝟏𝟐𝒊−𝟗𝒊² = 𝟏𝟐+𝒊−𝟔(−𝟏) 𝟏𝟔−𝟗(−𝟏) = 𝟏𝟐+𝒊+𝟔 𝟏𝟔+𝟗 = 𝟏𝟖+𝒊 𝟐𝟓 = 𝟏𝟖 𝟐𝟓 + 𝟏 𝟐𝟓 𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Write each quotient as a complex number. 13. 3𝑖 −2 + 𝑖 ∙ −𝟐−𝒊 −𝟐−𝒊 = 𝟑𝒊(−𝟐−𝒊) (−𝟐+𝒊)(−𝟐−𝒊) = −𝟔𝒊−𝟑𝒊² 𝟒+𝟐𝒊−𝟐𝒊−𝒊² = −𝟔𝒊−𝟑(−𝟏) 𝟒−(−𝟏) = −𝟔𝒊+𝟑 𝟒+𝟏 = 𝟑−𝟔𝒊 𝟓 = 𝟑 𝟓 − 𝟔 𝟓 𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Write each quotient as a complex number. 14. 7 5 − 2𝑖 ∙ 𝟓+𝟐𝒊 𝟓+𝟐𝒊 = 𝟕(𝟓+𝟐𝒊) (𝟓−𝟐𝒊)(𝟓+𝟐𝒊) = 𝟑𝟓+𝟏𝟒𝒊 𝟐𝟓+𝟏𝟎𝒊−𝟏𝟎𝒊−𝟒𝒊² = 𝟑𝟓+𝟏𝟒𝒊 𝟐𝟓−𝟒(−𝟏) = 𝟑𝟓+𝟏𝟒𝒊 𝟐𝟓+𝟒 = 𝟑𝟓+𝟏𝟒𝒊 𝟐𝟗 = 𝟑𝟓 𝟐𝟗 + 𝟏𝟒 𝟐𝟗 𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 15. 𝑥²+64=0 −𝟔𝟒 −𝟔𝟒 𝒙²=−𝟔𝟒 𝒙=± −𝟔𝟒 𝒙=±𝟖𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 16. 3𝑥²+27=0 −𝟐𝟕 −𝟐𝟕 𝟑𝒙²=−𝟐𝟕 𝟑 𝟑 𝒙²=−𝟗 𝒙=± −𝟗 𝒙=±𝟑𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 17. 𝑥²=−11 𝒙=± −𝟏𝟏 𝒙=±𝒊 𝟏𝟏

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 18. 2𝑥²+5=−31 −𝟓 −𝟓 𝟐𝒙²=−𝟑𝟔 𝟐 𝟐 𝒙²=−𝟏𝟖 𝒙=± −𝟏𝟖 𝒙=±𝟑𝒊 𝟐

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 19. 𝑥²+2𝑥+5=0 𝒂=𝟏 𝒃=𝟐 𝒄=𝟓 𝒙= −(𝟐)± (𝟐)²−𝟒(𝟏)(𝟓) 𝟐(𝟏) 𝒙= −𝟐± 𝟒−𝟐𝟎 𝟐 = −𝟐± −𝟏𝟔 𝟐 𝒙= −𝟐 𝟐 ± 𝟒 𝟐 𝒊 =−𝟏±𝟐𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 20. − 𝑥 2 +2𝑥−10=0 𝒂=−𝟏 𝒃=𝟐 𝒄=−𝟏𝟎 𝒙= −(𝟐)± (𝟐)²−𝟒(−𝟏)(−𝟏𝟎) 𝟐(−𝟏) 𝒙= −𝟐± 𝟒−𝟒𝟎 −𝟐 = −𝟐± −𝟑𝟔 −𝟐 𝒙= −𝟐 −𝟐 ± 𝟔 𝟐 𝒊 =𝟏±𝟑𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 21. 2𝑥²−3𝑥+5=0 𝒂=𝟐 𝒃=−𝟑 𝒄=𝟓 𝒙= −(−𝟑)± (−𝟑)²−𝟒(𝟐)(𝟓) 𝟐(𝟐) 𝒙= 𝟑± 𝟗−𝟒𝟎 𝟒 = 𝟑± −𝟑𝟏 𝟒 𝒙= 𝟑 𝟒 ± 𝟑𝟏 𝟒 𝒊 = 𝟑 𝟒 ± 𝟑𝟏 𝟒 𝒊

International Studies Charter School Mrs. Rivas International Studies Charter School Sec. 4-8 Complex numbers Solve each equation. 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 22. −4 𝑥 2 +6𝑥−3=0 𝒂=−𝟒 𝒃=𝟔 𝒄=−𝟑 𝒙= −(𝟔)± (𝟔)²−𝟒(−𝟒)(−𝟑) 𝟐(−𝟒) 𝒙= −𝟔± 𝟑𝟔−𝟒𝟖 −𝟖 = −𝟔± −𝟏𝟐 −𝟖 = 𝟑 𝟒 ± − 𝟑 𝟒 𝒊 𝒙= −𝟔 −𝟖 ± 𝟐 𝟑 −𝟖 𝒊