Warm-Up Solve Using Square Roots: 1.6x 2 = x 2 = 64

Section 4-6 Perform Operations with Complex Numbers

Vocabulary Imaginary Unit i – i = √-1, so i 2 = -1. Complex Number – A number a + bi where a and b are real numbers and i is the imaginary unit. Imaginary Number – A complex number a + bi where b ≠ 0. Complex Conjugates – Two complex numbers of the form a + bi and a – bi.

Sums and Differences of Complex Numbers Sum (a + bi) + (c + di) = (a + c) + (bi + di) Difference (a + bi) – (c + di) = (a – c) + (bi – di)

Example 1 Solve 2x = -72. – 18 – 18 2x 2 = -90 Step 1: Subtract 18 from both sides. √ x 2 = x = ± √-45 Step 2: Divide both sides by 2. Step 3: Take the square root of both sides. Step 4: Write in terms of i. x = ± i√45 Step 5: Simplify the radical. x = ± 3i√5 x = 3i√5 and – 3i√5

Example 2 Write the expression as a complex number in standard form. (a + bi) a.) (12 – 11i) + (-8 + 3i) b.) (15 – 9i) – (24 – 9i) c.) 35 – (13 + 4i) + i = 4 – 8i = (12 + (-8)) + ((-11i) + 3i) = (15 – 24) + ((-9i) – (-9i)) = – 9 = (35 – 13) + ((-4i) + 1i) = 22 – 3i

Example 3 Write the expression as a complex number in standard form. a.) -5i(8 – 9i) b.) (-8 + 2i)(4 – 7i) = – 40i + 45i 2 = (-5i)(8) – (-5i)(9i) = – i + 8i – 14i 2 = – i – 14(-1) = – 40i + 45(-1) = – 40i – 45 = – 45 – 40i = – i + 14 = – i

Example 4 Write the quotient 3 + 4i in standard form. 5 – i 3 + 4i 5 – i i + 20i + 4i 2 Step 1: Multiply by the conjugate 5 + i i – 5i – i i + 4(-1) 25 – (-1) Step 2: FOIL. Step 3: Simplify and use i 2 = -1. Step 4: Write in standard form. 5 + i i i 26 26

Homework Section 4-6 Pages 279 – 280 3, 5, 6, 8, 9, 10, 15 – 19, 21, 23 – 26, 29, 30, 32, 33, 51 – 53