WarmUp Exercises Find the exact value 2 144

Warm-Up Exercises Find the exact value. 2. –√ 144 1. √ 49 ANSWER 7 ANSWER – 12 3. Use calculator to approximate the value of 82 16 to the nearest tenth. ANSWER 2. 3

Warm-Up Exercises Homework Check

x x 2 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400

Warm-Up 1 Exercises EXAMPLE Use properties of square roots Simplify the expression. a. 4 4 = 81 81 b. 7 7 = 16 16 = 2 9 = 7 4

Warm-Up Exercises GUIDED PRACTICE 5. 9 64 6. 15 4 = 3 8 = 15 2 for Example 1 7. 8. 11 = 25 36 = 49 11 5 6 7

Warm-Up 3 Exercises EXAMPLE Solve a quadratic equation Solve 3 x 2 + 5 = 41 Write original equation. 3 x 2 = 36 Subtract 5 from each side. x 2 = 12 x = + Divide each side by 3. 12 The solutions are Take square roots of each side. 12 and – 12

Warm-Up 4 Exercises EXAMPLE Standardized Test Practice SOLUTION 1 (z + 3)2 = 7 5 (z + 3)2 = 35 z + 3 = + 35 z = – 3 + Write original equation. Multiply each side by 5. Take square roots of each side. 35 Subtract 3 from each side. The solutions are – 3 + 35 and – 35 The correct answer is C.

Warm-Up Exercises

Warm-Up Exercises

Warm-Up 1 Exercises EXAMPLE Solve a quadratic equation Solve 2 x 2 + 11 = – 37 Write original equation. 2 x 2 = – 48 x 2 = – 24 x = + – 24 x=+i 24 Subtract 11 from each side. Divide each side by 2. Take square roots of each side. Write in terms of i. ANSWER The solutions are i 24 and –i 24.

Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the equation. 1. x 2 = – 13. 3. 5 x 2 + 33 = 3. 2. x 2 + 11= 3.

Warm-Up Exercises GUIDED PRACTICE for Example 2 Write the expression as a complex number in standard form. 7. (9 – i) + (– 6 + 7 i) 8. (3 + 7 i) – (8 – 2 i) 9. – 4 – (1 + i) – (5 + 9 i) 9 – i – 6 + 7 i 3 + 7 i – 8 + 2 i – 4 – 1 – i – 5 – 9 i 3 + 6 i -5 + 9 i -10 – 10 i

Warm-Up 4 Exercises EXAMPLE Multiply complex numbers Write the expression as a complex number in standard form. a. 4 i(– 6 + i) = – 24 i + 4 i 2 = – 24 i + 4(– 1) = – 24 i – 4 = – 4 – 24 i Distributive property Use i 2 = – 1. Simplify. Write in standard form.

Warm-Up 4 Exercises EXAMPLE Multiply complex numbers b. (9 – 2 i)(– 4 + 7 i) = – 36 + 63 i + 8 i – 14 i 2 = – 36 + 71 i – 14(– 1) = – 36 + 71 i + 14 = – 22 + 71 i Multiply using FOIL. Simplify and use i 2 = – 1. Simplify. Write in standard form.

Warm-Up 5 Exercises EXAMPLE Divide complex numbers Write the quotient 7 + 5 i in standard form. 1 4 i 7 + 5 i = 7 + 5 i 1 – 4 i 1 + 4 i 2 7 + 28 i + 5 i + 20 i = 1 + 4 i – 16 i 2 7 + 33 i + 20(– 1) = 1 – 16(– 1) = – 13 + 33 i 17 13 + 33 i = – 17 17 Multiply numerator and denominator by 1 + 4 i, the complex conjugate of 1 – 4 i. Multiply using FOIL. Simplify and use i 2 = 1. Simplify. Write in standard form.

Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and 5 Write the expression as a complex number in standard form. 11. i(9 – i) 9 i – i 2 12. (3 + i)(5 – i) 15 – 3 i + 5 i – i 2 9 i – (-1) 15 + 2 i – (-1) 9 i + 1 15 + 2 i + 1 1 + 9 i 16 + 2 i

Warm-Up 6 Exercises EXAMPLE Plot complex numbers Plot the complex numbers in the same complex plane. a. 3 – 2 i b. – 2 + 4 i c. 3 i d. – 4 – 3 i SOLUTION a. To plot 3 – 2 i, start at the origin, move 3 units to the right, and then move 2 units down. b. To plot – 2 + 4 i, start at the origin, move 2 units to the left, and then move 4 units up. c. To plot 3 i, start at the origin and move 3 units up. d. To plot – 4 – 3 i, start at the origin, move 4 units to the left, and then move 3 units down.

Warm-Up 7 Exercises EXAMPLE Find absolute values of complex numbers Find the absolute value of (a) – 4 + 3 i and (b) – 3 i. a. – 4 + 3 i = (– 4)2+32 = 25 = 5 b. – 3 i = 0 + (– 3 i) = 02+ (– 3)2 = 9 = 3

Warm-Up Exercises GUIDED PRACTICE for Examples 6 and 7 Find the absolute value of: 15. 4 – i ANSWER 16. 17 – 3 – 4 i ANSWER 5 17. 2 + 5 i ANSWER 29 18. – 4 i ANSWER 4

Warm-Up Exercises Classwork Assignment: WS 4. 5 (13 -27 odd) and WS 4. 6 (1 -40 multiples of 3)