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Class Notes and Examples Complex Numbers and Roots An imaginary number is the square root of a negative number. Use the definition 1 i to simplify square roots. Simplify. 25 25 1 Factor out 1. 25 Separate roots. 1 5 1 Simplify. 5i Express in terms of i. 48 48 1 Factor out 1. 48 Separate roots. 1 16 3 1 Factor the perfect square. 4 3 1 Simplify. 4i 3 Express in terms of i. Real Imaginary Complex numbers are numbers that can be written in the form a bi. The complex conjugate of a bi is a bi. The complex conjugate of 5i is 5i. Write as a bi Find 0 5i 5i Express each number in terms of i. 1. 72 36 2 1 ________________________ 4. 5 54 ________________________ 2. 4 45 4 3. 100 9 5 1 ________________________ 5. 2 64 ________________________ ________________________ 6. 98 ________________________ Find each complex conjugate. 7. 9i ________________________ 8. 1 4i ________________________ 9. 12 i ________________________ Class Notes and Examples Complex Numbers and Roots (continued) You can use the square root property and imaginary solutions. 1 i to solve quadratic equations with Solve x2 64. x 2 64 Take the square root of both sides. x 8i Express in terms of i. 2 Check each root: 8i 64i 641 64 2 Remember: 1 2 i2 1 8i2 64i 2 641 64 Solve 5x2 80 0. 5x2 80 Subtract 80 from both sides. x2 16 Divide both sides by 5. x 2 16 Take the square root of both sides. x 4i Express in terms of i. Check each root: 54i 2 80 54i 2 80 516i 2 80 516i 2 80 801 80 801 80 0 0 Solve each equation. 10. x2 18 0 x2 18 x 9 2 1 ________________________ 13. x 100 0 2 11. 6x2 24 0 12. x2 49 0 6x2 24 ____________________ ____________________ ____________________ ________________________ 14. 3x 108 0 2 ________________________ 15. x2 12 0 ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ Class Notes and Examples Operations with Complex Numbers (continued) To add or subtract complex numbers, add the real parts and then add the imaginary parts. First, group to add the real parts and 3 2i 4 5i the imaginary parts. This is similar to 3 4 2i 5i adding like terms. 7 3i Remember to distribute when subtracting. Then group to add the real parts and the imaginary parts. 4 i 2 6i 4 i 2 6i 4 2 i 6i 6 7i Use the Distributive Property to multiply complex numbers. Remember that i 2 1. 3i 2 i 6i 3i 2 Distribute. 6i 31 Use i 2 1. 3 6i Write in the form a bi. 4 2i 5 i 20 4i 10i 2i 2 Multiply. 20 6i 21 Combine imaginary parts and use i 2 1. 22 6i Combine real parts. Add, subtract, or multiply. Write the result in the form a bi. 12. 6 i 3 2i 6 3 i 2i ________________________ 15. 2 4i 1 4i ________________________ 18. 6 5i 5i 6 ________________________ 13. 9 3i 2 i 9 3i 2 i ________________________ 16. 1 7i 1 5i ________________________ 19. 2 i 3i 2 ________________________ 14. 3 i 2 2i 6 6i 2i 2i2 ________________________ 17. 5i 4 3i ________________________ 20. 2 4i 2 ________________________