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Complex Numbers Mastery: Operations, Equations, and Applications

Dive deep into complex numbers with lessons on square roots, pure imaginary numbers, equations with imaginary solutions, and real-world examples. Ace the 5-minute checks and enhance your understanding of AC circuits.

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Complex Numbers Mastery: Operations, Equations, and Applications

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–3) CCSS Then/Now New Vocabulary Example 1: Square Roots of Negative Numbers Example 2: Products of Pure Imaginary Numbers Example 3: Equation with Pure Imaginary Solutions Key Concept: Complex Numbers Example 4: Equate Complex Numbers Example 5: Add and Subtract Complex Numbers Example 6: Real-World Example: Multiply Complex Numbers Example 7: Divide Complex Numbers Lesson Menu

  3. Solve x2 – x = 2 by factoring. A. 2, –1 B. 1, 2 C. 1, 1 D. –1, 1 5-Minute Check 1

  4. Solve c2 – 16c + 64 = 0 by factoring. A. 2 B. 4 C. 8 D. 12 5-Minute Check 2

  5. Solve z2 = 16z by factoring. A. 1, 4 B. 0, 16 C. –1, 4 D. –16 5-Minute Check 3

  6. A. B.0 C.–1 D. –1 Solve 2x2 + 5x + 3 = 0 by factoring. 5-Minute Check 4

  7. Write a quadratic equation with the roots –1 and 6 in the form ax2 + bx + c, where a, b, and c are integers. A.x2 – x + 6 = 0 B.x2 + x + 6 = 0 C.x2 – 5x – 6 = 0 D.x2 – 6x + 1 = 0 5-Minute Check 5

  8. In a rectangle, the length is three inches greater than the width. The area of the rectangle is 108 square inches. Find the width of the rectangle. A. 6 in. B. 8 in. C. 9 in. D. 12 in. 5-Minute Check 6

  9. Content Standards N.CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. N.CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Mathematical Practices 6 Attend to precision. CCSS

  10. You simplified square roots. • Perform operations with pure imaginary numbers. • Perform operations with complex numbers. Then/Now

  11. imaginary unit • pure imaginary number • complex number • complex conjugates Vocabulary

  12. A. Square Roots of Negative Numbers Answer: Example 1

  13. B. Answer: Square Roots of Negative Numbers Example 1

  14. A. B. C. D. A. Example 1

  15. A. B. C. D. B. Example 1

  16. Products of Pure Imaginary Numbers A.Simplify –3i ● 2i. –3i ● 2i= –6i2 = –6(–1) i2 = –1 = 6 Answer: 6 Example 2

  17. B. Answer: Products of Pure Imaginary Numbers Example 2

  18. A. Simplify 3i ● 5i. A. 15 B. –15 C. 15i D. –8 Example 2

  19. B. Simplify . A. B. C. D. Example 2

  20. Equation with Pure Imaginary Solutions Solve 5y2 + 20 = 0. 5y2 + 20 = 0 Original equation 5y2 = –20 Subtract 20 from each side. y2 = –4 Divide each side by 5. Take the square root of each side. Answer:y = ±2i Example 3

  21. Solve 2x2 + 50 = 0. A.±5i B.±25i C.±5 D.±25 Example 3

  22. Concept

  23. Equate Complex Numbers Find the values of x and y that make the equation 2x + yi = –14 – 3i true. Set the real parts equal to each other and the imaginary parts equal to each other. 2x = –14 Real parts x = –7 Divide each side by 2. y = –3 Imaginary parts Answer:x = –7, y = –3 Example 4

  24. Find the values of x and y that make the equation 3x – yi = 15 + 2i true. A.x = 15y = 2 B.x = 5y = 2 C.x = 15y = –2 D.x = 5y = –2 Example 4

  25. Add and Subtract Complex Numbers A.Simplify (3 + 5i) + (2 – 4i). (3 + 5i) + (2 – 4i) = (3 + 2) + (5 – 4)iCommutative and Associative Properties = 5 + i Simplify. Answer: 5 + i Example 5

  26. Add and Subtract Complex Numbers B.Simplify (4 – 6i) – (3 – 7i). (4 – 6i) – (3 – 7i) = (4 – 3) + (–6 + 7)iCommutative and Associative Properties = 1 + i Simplify. Answer: 1 + i Example 5

  27. A.Simplify (2 + 6i) + (3 + 4i). A. –1 + 2i B. 8 + 7i C. 6 + 12i D. 5 + 10i Example 5

  28. B.Simplify (3 + 2i) – (–2 + 5i). A. 1 + 7i B. 5 – 3i C. 5 + 8i D. 1 – 3i Example 5

  29. Multiply Complex Numbers ELECTRICITYIn an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I ● Z. Find the voltage in a circuit with current 1 + 4j amps and impedance 3 – 6j ohms. E = I ● Z Electricity formula = (1 + 4j)(3 – 6j) I = 1 + 4j, Z = 3 – 6j = 1(3) + 1(–6j) + 4j(3) + 4j(–6j) FOIL = 3 – 6j + 12j – 24j2 Multiply. = 3 + 6j – 24(–1) j2 = –1 = 27 + 6jAdd. Answer: The voltage is 27 + 6j volts. Example 6

  30. ELECTRICITYIn an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I ● Z. Find the voltage in a circuit with current 1 – 3j amps and impedance 3 + 2j ohms. A. 4 – j B. 9 – 7j C. –2 – 5j D. 9 – j Example 6

  31. A. Answer: Divide Complex Numbers 3 – 2i and 3 + 2i are conjugates. Multiply. i2 = –1 a + bi form Example 7

  32. B. Multiply by . Answer: Divide Complex Numbers Multiply. i2 = –1 a + bi form Example 7

  33. A. A. B.3 + 3i C.1 + i D. Example 7

  34. B. A. B. C. D. Example 7

  35. End of the Lesson

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