Warm-Up Exercises

Find the exact value.

ANSWER 7 ANSWER –12

1. √49 2. –√144

ANSWER 2.3

163. Use calculator to approximate the value of

to the nearest tenth.

82

Warm-Up ExercisesHomework CheckHomework Check

x x2

1 1

2 4

3 9

4 16

5 25

6 36

7 49

8 64

9 81

10 100

x x2

11 121

12 144

13 169

14 196

15 225

16 256

17 289

18 324

19 361

20 400

Warm-Up ExercisesEXAMPLE 1 Use properties of square roots

Simplify the expression.

a.

481 =

4

81=

2 9

b.

716 =

7

16=

47

Warm-Up ExercisesGUIDED PRACTICEGUIDED PRACTICE for Example 1

5. 3 8

964

6. 154

215=

= 1125

8. 3649

511

76

7. =

=

Warm-Up ExercisesEXAMPLE 3 Solve a quadratic equation

Solve 3x2 + 5 = 41.

3x2 + 5 = 41 Write original equation.

3x2 = 36 Subtract 5 from each side.

x2 = 12 Divide each side by 3.

x =

+ 12 Take square roots of each side.

The solutions are and 12 12–

Warm-Up ExercisesEXAMPLE 4 Standardized Test Practice

SOLUTION

15

(z + 3)2 = 7 Write original equation.

(z + 3)2 = 35 Multiply each side by 5.

z + 3 = + 35 Take square roots of each side.

z = –3 + 35 Subtract 3 from each side.

The solutions are –3 + and –3 – 35 35

The correct answer is C.

Warm-Up Exercises

Warm-Up Exercises

i 1

9 91 91 3i i3

15 151 151 15i 15i

24 241 241 24i

12 i

Warm-Up ExercisesEXAMPLE 1 Solve a quadratic equation

Solve 2x2 + 11 = –37.

2x2 + 11 = –37 Write original equation.

2x2 = –48 Subtract 11 from each side.

x2 = –24 Divide each side by 2.

Take square roots of each side.x = + –24

Write in terms of i.x = + i 24

ANSWER

The solutions are i 24 and –i 24 .

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Solve the equation.

x2 = –13.1. x2 + 11= 3.2.

5x2 + 33 = 3 .3.

Warm-Up ExercisesGUIDED PRACTICE for Example 2

Write the expression as a complex number in standard form.

7. (9 – i) + (–6 + 7i) 8. (3 + 7i) – (8 – 2i) 9. –4 – (1 + i) – (5 + 9i)

9 – i – 6 + 7i

3 + 6i

3 + 7i – 8 + 2i

-5 + 9i

–4 – 1 – i – 5 – 9i

-10 – 10i

Warm-Up ExercisesEXAMPLE 4 Multiply complex numbers

Write the expression as a complex number in standardform.

a. 4i(–6 + i)

a. 4i(–6 + i) = –24i + 4i2 Distributive property

= –24i + 4(–1) Use i2 = –1.

= –24i – 4 Simplify.

= –4 – 24i Write in standard form.

1 :Recall 2 i

Warm-Up ExercisesEXAMPLE 4 Multiply complex numbers

b. (9 – 2i)(–4 + 7i)

Multiply using FOIL.= –36 + 63i + 8i – 14i2

= –36 + 71i – 14(–1) Simplify and use i2 = – 1 .

= –36 + 71i + 14 Simplify.

= –22 + 71i Write in standard form.

Warm-Up ExercisesEXAMPLE 5 Divide complex numbers

Write the quotient in standard form.

7 + 5i 1 4i

7 + 5i 1 – 4i

7 + 5i 1 – 4i= 1 + 4i

1 + 4i Multiply numerator and denominator by 1 + 4i, the complex conjugate of 1 – 4i.

7 + 28i + 5i + 20i2

1 + 4i – 4i – 16i2= Multiply using FOIL.

7 + 33i + 20(–1)1 – 16(–1)= Simplify and use i2 = 1.

–13 + 33i 17= Simplify.

1317 –= + 33

17 i Write in standard form.

Warm-Up ExercisesGUIDED PRACTICE for Examples 3, 4 and 5

11.

1 + 9i

i(9 – i) 12. (3 + i)(5 – i)

16 + 2i

Write the expression as a complex number in standard form.

9i – i2

9i – (-1)

9i + 1

15 – 3i + 5i – i2

15 + 2i – (-1)

15 + 2i + 1

Warm-Up ExercisesEXAMPLE 6 Plot complex numbers

Plot the complex numbers in the same complex plane.

a. 3 – 2i b. –2 + 4i c. 3i d. –4 – 3i

SOLUTION

a. To plot 3 – 2i, start at the origin, move 3 units to the right, and then move 2 units down.

b. To plot –2 + 4i, start at the origin, move 2 units to the left, and then move 4 units up.

c. To plot 3i, start at the origin and move 3 units up.

d. To plot –4 – 3i, start at the origin, move 4 units to the left, and then move 3 units down.

Warm-Up ExercisesEXAMPLE 7 Find absolute values of complex numbers

Find the absolute value of (a) –4 + 3i and (b) –3i.

a. –4 + 3i = (–4)2+32 = 25 = 5

b. –3i = 02+ (–3)2 = 9 = 30 + (–3i) =

Warm-Up ExercisesGUIDED PRACTICE for Examples 6 and 7

15. 4 – i

–3 – 4i

5

16.

18. –4i

4

17. 2 + 5i

29

17ANSWER

ANSWER

ANSWER

ANSWER

Find the absolute value of:

Warm-Up Exercises

Classwork Assignment:WS 4.5 (13-27 odd) and WS 4.6 (1-40 multiples of 3)