Warm-Up Exercises

1. 52 ANSWER 132

2. 15

3ANSWER

5

15

5–+ANSWER

ANSWER 5,– 9–( )27+n4 16=4.

3x 2 8 23+ =3.

Solve the equation.

5. An object falls off a 1000 foot tall building. How long does it take to hit the ground?

Warm-Up Exercises

ANSWER about 7.9 sec

Example 1 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 +– 7i

Example 1 Solve a Quadratic Equation

Write original equation. b. =3x 2 29–5–

Add 5 to each side.=3x 2 24–

Divide each side by 3.=x 2 8–

Write in terms of i.=x +– 8i

Take the square root of each side. =x +– 8–

Simplify the radical. =x +– 2i2

Checkpoint

Solve the equation.

Solve a Quadratic Equation

1. =x 2 3– ANSWER 3,i 3i–

2. =x 2 7–

3. =x 2 20–

4. =x 2 3 2+ –

5. =y 2 4– 12–

ANSWER 7,i 7i–

ANSWER 5,2 52–i i

ANSWER 5,i 5i–

ANSWER 2,2 22–i i

Checkpoint

Solve the equation.

Solve a Quadratic Equation

6. =2x 2 7 17+ – ANSWER 3,2 32–i i

Example 2 Add Complex Numbers

Write as a complex number in standard form.

2i3(

(

+ i1(

(

–+

SOLUTION

Group real and imaginary terms.

2i3(

(

+ i1(

(

–+ = 13( +

(

12(

(

i+ –

Write in standard form.

= 4 + i

Example 3 Subtract Complex Numbers

Write as a complex number in standard form.

Simplify.= 5 + 0i

2i6(

(

– – 2i1(

(

–

SOLUTION

Group real and imaginary terms.

2i6(

(

= 16(

(

22(

(

i+ +– – 2i1(

(

– – –

Write in standardform.

= 5

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

7. 2i4(

(

– + 3i1(

(

+ ANSWER i5 +

8. i3((

– + 4i2((

+ ANSWER 3i5 +

ANSWER 3i2 +9. 6i4(

(

+ 3i2(

(

+–

10. 4i2(

(

+ 7i2(

(

+–– ANSWER 3i4– –

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

11. 2i1(

(

– + 5i4(

(

+ ANSWER 3i5 +

ANSWER 3i3 +12. i2((

– ((

– 4i1– –

Example 4 Multiply Complex Numbers

Write the expression as a complex number in standard form.

a. b.1( 3i

(

+–2i 3i6(

(

+ 3i4(

(

–

SOLUTION

Multiply using distributive property.

1( 3i

(

+–2i = 2i 6i 2– +a.

1

(

(–2i 6–= + Use i 2 1.= –

6 2i––= Write in standard form.

Example 4 Multiply Complex Numbers

Multiply using FOIL. b. 3i6(

(

+ 3i4(

(

– 24 18i– 12i+ 9i 2–=

24 6i– – 9i 2= Simplify.

24 6i– – 1

(

(–9= Use i 2 1.= –

6i33 –= Write in standard form.

SOLUTION

2i3 +

2i1 –

2i3 +

2i1 –

2i1 +

2i1 += •

Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i.

+–

Example 5 Divide Complex Numbers

Write as a complex number in standard form.2i3 +

2i1 –

Multiply using FOIL. 1

2i3 6i+ – 4i 2+

2i2i+ – 4i 2–=

3 8i+ 1

(

(–4+

1 – 1

(

(–4= Simplify and use i 2 1.= –

Example 5 Divide Complex Numbers

8i+–1

5= Simplify.

51–

58 i+= Write in standard form.

Checkpoint

Write the expression as a complex number in standard form.

Multiply and Divide Complex Numbers

13. i2(

(

–3i ANSWER 6i3 +

14. ( 2i1(

+ i2((

– ANSWER 3i4 +

15.i2 +

i1 –ANSWER

2

1+2

3i