Warm-Up Exercises 1. 52 ANSWER 13 2 2. 15 3 ANSWER 5 15 3x 2 8 23 + =

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Warm-Up Exercises 1. 52 ANSWER 13 2 2. 15 3 ANSWER 5 15 3x 2 8 23 + = 3. Solve the equation. 5 – + ANSWER ( )2 7 + n 4 16 = 4. ANSWER 5, – 9

Warm-Up Exercises An object falls off a 1000 foot tall building. How long does it take to hit the ground? ANSWER about 7.9 sec

Example 1 Solve the equation. = 7x 2 49 – a. b. = 3x 2 5 – 29 SOLUTION Solve a Quadratic Equation Solve the equation. = 7x 2 49 – a. b. = 3x 2 5 – 29 SOLUTION Write original equation. = 7x 2 49 – a. Divide each side by 7. = x 2 7 – Take the square root of each side. = x + – 7 Write in terms of i. = x + – 7 i

Example 1 b. = 3x 2 29 – 5 = 3x 2 24 – = x 2 8 – = x + – 8 = x + – 8 i Solve a Quadratic Equation b. = 3x 2 29 – 5 Write original equation. Add 5 to each side. = 3x 2 24 – Divide each side by 3. = x 2 8 – Take the square root of each side. = x + – 8 Write in terms of i. = x + – 8 i Simplify the radical. = x + – 2 i 4

Checkpoint Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7 Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7 – ANSWER 7, i 7 – 3. = x 2 20 – ANSWER 5, 2 5 – i 4. = x 2 3 2 + – ANSWER 5, i 5 – 5. = y 2 4 – 12 ANSWER 2, 2 – i

Checkpoint Solve the equation. 6. = 2x 2 7 17 + – ANSWER 3, 2 3 – i Solve a Quadratic Equation Solve the equation. 6. = 2x 2 7 17 + – ANSWER 3, 2 3 – i

Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i ( Example 2 Add Complex Numbers Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i ( SOLUTION Group real and imaginary terms. 2i 3 ( + i 1 – = 2 Write in standard form. = 4 + i 7

Write as a complex number in standard form. 2i 6 ( – 1 Example 3 Subtract Complex Numbers Write as a complex number in standard form. 2i 6 ( – 1 SOLUTION Group real and imaginary terms. 2i 6 ( = 1 2 i + – Simplify. = 5 + 0i Write in standard form. = 5 8

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 7. ( 4 – ( 2i ( + 1 + 3i ( ANSWER i 5 + 8. i 3 ( – + 4i 2 ANSWER 3i 5 + 9. 6i 4 ( + 3i 2 – ANSWER 3i 2 + 10. 4i 2 ( + 7i – ANSWER 3i 4 –

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 11. 2i 1 ( – + 5i 4 ANSWER 3i 5 + 12. i 2 ( – 4i 1 ANSWER 3i 3 +

Write the expression as a complex number in standard form. Example 4 Multiply Complex Numbers Write the expression as a complex number in standard form. a. 1 ( 3i + – 2i b. 3i 6 ( + 3i 4 ( – SOLUTION Multiply using distributive property. 1 ( 3i + – 2i = 6i 2 a. 1 ( – 2i 6 = + Use i 2 1. 6 2i – = Write in standard form.

Example 4 b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = 24 6i – 9i 2 = 24 6i – 1 ( Multiply Complex Numbers b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = Multiply using FOIL. 24 6i – 9i 2 = Simplify. 24 6i – 1 ( 9 = Use i 2 1. 6i 33 – = Write in standard form. 12

Write as a complex number in standard form. 2i 3 + 1 – Example 5 Divide Complex Numbers Write as a complex number in standard form. 2i 3 + 1 – SOLUTION 2i 3 + 1 – = • Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i. Multiply using FOIL. 1 2i 3 6i + – 4i 2 = 3 8i + 1 ( – 4 = Simplify and use i 2 1. 13

Example 5 8i + – 1 5 = 5 1 – 8 i + = Divide Complex Numbers Simplify. Write in standard form. 14

Write the expression as a complex number in standard form. Checkpoint Multiply and Divide Complex Numbers Write the expression as a complex number in standard form. 13. i 2 ( – 3i ANSWER 6i 3 + 14. ( 2i 1 + i 2 – ANSWER 3i 4 + 15. i 2 + 1 – ANSWER 2 1 + 3 i