Beginning Algebra, 11th Edition

590 CHAPTER 9 Quadratic Equations

Subtraction
To subtract complex numbers, change the number
following the subtraction sign to its negative and add.

Multiplication
Multiply complex numbers in the same way polynomials
are multiplied. Replace with

Division
Divide complex numbers by multiplying the numerator
and the denominator by the conjugate of the denominator.

Complex Solutions
A quadratic equation may have nonreal complex
solutions. This occurs when the discriminant is negative.
The quadratic formula will give complex solutions in
such cases.

i^2 - 1.

Subtract.

Definition of

subtraction

Add real parts and

add imaginary parts.

Multiply.

FOIL method

Multiply.

Multiply.

Add.

Divide.

is the conjugate

of

Multiply.

Multiply;

Subtract.

Standard form

Solve for all complex solutions of

Simplify.

,or

The solution set is e-

1

2



23

2

if.

• 
  

1

2



i 23

2

x=

• 1 i 23
  2

x=

• 1  2 - 3
  2

x=

• 1  21 - 4
  2

x= a=1,b=1,c= 1

• 1  212 - 4112112
  2112

x^2 +x+ 1 =0.

=

12

37

• 2
  37

i

=

12 - 2 i

37

= i^2 =- 1

12 - 2 i

36 - 1 - 12

=

216 - i 2

36 - i^2

6 +i.

=^26 - i

6 +i

#^6 - i

6 - i

2

6 +i

= 25 - 25 i

= 21 - 25 i+ 4

= 21 - 25 i- 41 - 12 i^2 =- 1

= 21 - 28 i+ 3 i- 4 i^2

= 7132 + 71 - 4 i 2 +i 132 +i 1 - 4 i 2

17 +i 213 - 4 i 2

= 3 + 8 i

= 15 - 22 + 14 + 42 i

= 15 + 4 i 2 + 1 - 2 + 4 i 2

15 + 4 i 2 - 12 - 4 i 2

CONCEPTS EXAMPLES

(continued)

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