Chapter 1.6. Perform Operations with Complex Numbers
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1 Chapter 1.6 Perform Operations with Complex Numbers
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3 EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x = 37. 2x = 37 2x 2 = 48 Write original equation. Subtract 11 from each side. x 2 = 24 Divide each side by 2. x = + 24 x = + i 24 Take square roots of each side. Write in terms of i. ANSWER x = + 2i 6 Simplify radical. The solutions are 2i 6 and 2i 6.
4 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 1. x 2 = 13. x 2 = 13. x = + 13 x = + i 13 x = + i 13 Write original equation. Take square roots of each side. Write in terms of i. Simplify radical. ANSWER The solutions are x =i 13 and i 13.
5 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 2. x 2 = 38. x 2 = 38. x = + 38 x = + i 38 x = + i 38 Write original equation. Take square roots of each side. Write in terms of i. Simplify radical. ANSWER The solutions are x =i 38 and i 38.
6 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 3. x = 3. x = 3. x 2 = 8. x = + 8 x = + i 8 x = + 2i 2 Write original equation. Subtract 11 from each side. Take square roots of each side. Write in terms of i. Simplify radical. ANSWER The solutions are 2i 2 and 2i 2.
7 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 4. x 2 8 = 36. x 2 8 = 36. Write original equation. x 2 = 28. x = + 28 x = + i 28 x = + 2i 7 ANSWER Add 8 to each side. Take square roots of each side. Write in terms of i. Simplify radical. The solutions are 2i 7 and 2i 7.
8 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 5. 3x 2 7 = 31. 3x 2 7 = 31. 3x 2 = 24. x 2 = 8. ANSWER x = + 8 x = + i 8 Write original equation. Add 7 to each side. Divided each side by 3 Take square roots of each side. Write in terms of i. x = + 2i 2 Simplify radical. The solutions are 2i 2 and 2i 2.
9 Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 6. 5x = 3. 5x = 3. 5x 2 = 30. x 2 = 6. ANSWER x = + 6 x = + i 6 x = + i 6 Write original equation. Add 7 to each side. Divided each side by 3 Take square roots of each side. Write in terms of i. Simplify radical. The solutions are i 6 and i 6.
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12 EXAMPLE Warm-Up 2 Exercises Add and subtract complex numbers Write the expression as a complex number in standard form. a. (8 i) + (5 + 4i) b. (7 6i) (3 6i) c. 10 (6 + 7i) + 4i SOLUTION a. (8 i) + (5 + 4i) = (8 + 5) + ( 1 + 4)i Definition of complex addition = i Write in standard form. b. (7 6i) (3 6i) = (7 3) + ( 6 + 6)i = 4 + 0i Simplify. Definition of complex subtraction = 4 Write in standard form.
13 EXAMPLE Warm-Up 2 Exercises Add and subtract complex numbers c. 10 (6 + 7i) + 4i = [(10 6) 7i] + 4i = (4 7i) + 4i Simplify. Definition of complex subtraction = 4 + ( 7 + 4)i Definition of complex addition = 4 3i Write in standard form.
14 Warm-Up Exercises GUIDED PRACTICE for Example 2 Write the expression as a complex number in standard form. 7. (9 i) + ( 6 + 7i) = (9 i) + ( 6 + 7i) = (9 6) + ( 1 + 7)i Definition of complex addition = 3 + 6i Write in standard form.
15 Warm-Up Exercises GUIDED PRACTICE for Example 2 Write the expression as a complex number in standard form. 8. (3 + 7i) (8 2i) = (3 + 7i) (8 2i) = (3 8) + (7 + 2)i Definition of complex subtraction = 5 + 9i Write in standard form.
16 Warm-Up Exercises GUIDED PRACTICE for Example 2 Write the expression as a complex number in standard form (1 + i) (5 + 9i) = 4 (1 + i) (5 + 9i) = [( 4 1 5) i] 9i Definition of complex subtraction = ( 10 i) 9i Simplify. = 10 + ( 1 9)i Definition of complex addition = 10 10i Write in standard form.
17 MULTIPLYING COMPLEX NUMBERS To multiply two complex numbers, use the distributive property or the FOIL method just as you do when multiplying real numbers or algebraic expressions
18 EXAMPLE Warm-Up 4 Exercises Multiply complex numbers Write the expression as a complex number in standard form. a. 4i( 6 + i) b. (9 2i)( 4 + 7i) SOLUTION a. 4i( 6 + i) = 24i + 4i 2 Distributive property = 24i + 4( 1) Use i 2 = 1. = 24i 4 Simplify. = 4 24i Write in standard form.
19 EXAMPLE Warm-Up 4 Exercises Multiply complex numbers b. (9 2i)( 4 + 7i) Multiply using FOIL. = i + 8i 14i 2 = i 14( 1) Simplify and use i 2 = 1. = i + 14 Simplify. = i Write in standard form.
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21 EXAMPLE Warm-Up 5 Exercises Divide complex numbers Write the quotient form i 1 4i = 7 + 5i 1 4i 1 + 4i 1 + 4i 7 + 5i 1 4i in standard Multiply numerator and denominator by 1 + 4i, the complex conjugate of 1 4i. = = = i + 5i + 20i i 4i 16i i + 20( 1) 1 16( 1) i 17 Multiply using FOIL. Simplify and use i 2 = 1. Simplify.
22 EXAMPLE Warm-Up 5 Exercises Divide complex numbers = i Write in standard form.
23 Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and i(9 i) SOLUTION i(9 i) = 9i i 2 Distributive property = 9i + ( 1) 2 Use i 2 = 1. = 9i + 1 Simplify. = 1 + 9i Write in standard form.
24 Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and (3 + i) (5 i) = 15 3i + 5i i 2 Multiply using FOIL. = 15 3i + 5i (1) 2 Simplify and use i 2 = 1. = 15 3i + 5i + 1 Simplify. = i Write in standard form.
25 Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and i i = = = = 5 1 i 1 + i 1 i 5 5i 1 i + i i 2 5 5i i 2 Multiply numerator and denominator by 1 i, the complex conjugate of 1 + i. Multiply using FOIL. Simplify and use i 2 = 1. Simplify.
26 GUIDED Warm-Up PRACTICE Exercises for Examples 3, 4 and 5 = i Write in standard form.
27 Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and i 3 2i 5 + 2i 3 2i = 5 + 2i 3 2i 3 + 2i 3 + 2i Multiply numerator and denominator 3 + 2i, the complex conjugate of 3 2i. = = = i + 6i + 4i i 6i 4i i + 4( 1) 9 4( 1) i 13 Multiply using FOIL. Simplify and use i 2 = 1. Simplify.
28 Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and 5 = i Write in standard form.
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