4-8 Complex Numbers
Quick Review
A complex number is written in the form a + bi, where a and b are real numbers, and i is equal to
−
1
.
square root of negative 1 end root , . If
b
=
0
,
b equals 0 comma a + bi is a real number. If
b
≠
0
,
b not equal to 0 comma a + bi is an imaginary number. You can use the Quadratic Formula or completing the square to find the imaginary solutions of quadratic equations.
Example
Use the Quadratic Formula to solve
3
x
2
−
4
x
+
2
=
0
.
3 x squared , minus 4 x plus 2 equals 0 .
x
=
−
(
−
4
)
±
(
−
4
)
2
−
4
(
3
)
(
2
)
2
(
3
)
Enter
a
,
b
,
and
c
into the
quadratic formula
.
x
=
4
±
16
−
24
6
=
4
±
−
8
6
Simplify
.
x
=
2
3
±
2
3
i
Write the solutions
.
table with 3 rows and 3 columns , row1 column 1 , x , column 2 equals . fraction negative . open , negative 4 , close . plus minus . square root of open , negative 4 , close squared . minus 4 , open 3 close . open 2 close end root , over 2 , open 3 close end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap enter , eh comma b comma , and , c . intothe , row2 column 1 , quadraticformula . . , end table , row2 column 1 , x , column 2 equals . fraction 4 plus minus . square root of 16 minus 24 end root , over 6 end fraction . equals . fraction 4 plus minus , square root of negative 8 end root , over 6 end fraction , column 3 cap simplify , . , row3 column 1 , x , column 2 equals , 2 thirds , plus minus , fraction square root of 2 , over 3 end fraction , i , column 3 cap writethesolutions . . , end table
Exercises
Simplify each expression using the imaginary unit i.
-
−
24
square root of negative 24 end root
-
−
2
−
3
square root of negative 2 end root , minus 3
-
(
4
+
−
25
)
(
−
100
)
open . 4 plus , square root of negative 25 end root . close . open , square root of negative 100 end root , close
-
2
−
24
+
6
2 , square root of negative 24 end root , plus 6
Simplify each expression.
-
(
9
+
7
i
)
−
(
6
−
2
i
)
open 9 plus 7 i close minus open 6 minus 2 i close
-
(
3
+
11
i
)
+
(
10
+
9
i
)
open 3 plus 11 i close plus open 10 plus 9 i close
-
(
1
−
9
i
)
(
3
+
2
i
)
open 1 minus 9 i close open 3 plus 2 i close
-
(
3
i
)
2
−
3
(
1
+
5
i
)
open 3 i , close squared , minus 3 open 1 plus 5 i close
-
4
−
6
i
2
i
fraction 4 minus 6 i , over 2 i end fraction
-
2
−
3
i
1
+
5
i
fraction 2 minus 3 i , over 1 plus 5 i end fraction
Solve each equation.
-
x
2
+
9
=
0
x squared , plus 9 equals 0
-
5
x
2
−
2
x
+
1
=
0
5 x squared , minus 2 x plus 1 equals 0
-
−
x
2
+
4
x
=
10
negative , x squared , plus 4 x equals 10
-
7
x
2
+
8
x
=
−
6
7 x squared , plus 8 x equals negative 6
4-9 Quadratic Systems
Quick Review
A system of quadratic equations can be solved by substitution or by graphing. You can use these methods to solve a linear-quadratic system or a quadratic-quadratic system. Use graphing to solve a quadratic system of inequalities.
Example
Use substitution to solve
{
y
=
2
x
2
+
2
x
−
10
y
=
x
2
+
5
x
−
6
.
left brace . table with 2 rows and 1 column , row1 column 1 , y equals 2 , x squared , plus 2 x minus 10 , row2 column 1 , y equals , x squared , plus 5 x minus 6 , end table . .
|
2
x
2
+
2
x
−
10
=
x
2
+
5
x
−
6
2 x squared , plus 2 x minus 10 equals , x squared , plus 5 x minus 6
|
Substitute for y. |
|
x
2
−
3
x
−
4
=
0
x squared , minus 3 x minus 4 equals 0
|
Rewrite in standard form. |
|
(
x
+
1
)
(
x
−
4
)
=
0
open x plus 1 close open x minus 4 close equals 0
|
Factor. |
|
x
=
−
1
or
x
=
4
x equals negative 1 , or , x equals 4
|
Solve for x. |
|
y
=
(
−
1
)
2
+
5
(
−
1
)
−
6
=
−
10
y equals open negative 1 , close squared , plus 5 open negative 1 close minus 6 equals negative 10
|
Substitute for x then solve for y.
|
|
y
=
(
4
)
2
+
5
(
4
)
−
6
=
30
y equals open 4 , close squared , plus 5 open 4 close minus 6 equals 30
|
|
|
(
−
2
,
−
10
)
open negative 2 comma negative 10 close and (4, 30) |
Write solutions as ordered pairs. |
Exercises
Solve each system by substitution.
-
{
y
=
x
2
−
7
x
−
6
y
=
8
−
2
x
left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 7 x minus 6 , row2 column 1 , y equals 8 minus 2 x , end table
-
{
y
=
−
x
2
−
2
x
+
8
y
=
x
2
−
8
x
−
12
left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 2 x plus 8 , row2 column 1 , y equals , x squared , minus 8 x minus 12 , end table
Solve each system by graphing.
-
{
y
=
−
x
2
−
10
x
+
12
y
=
x
2
−
6
x
−
18
left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 10 x plus 12 , row2 column 1 , y equals , x squared , minus 6 x minus 18 , end table
-
{
y
=
x
2
−
x
−
18
y
=
2
x
+
3
left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus x minus 18 , row2 column 1 , y equals 2 x plus 3 , end table
Solve each system of inequalities.
-
{
y
<
x
+
4
y
≥
x
2
+
2
x
+
2
left brace . table with 2 rows and 1 column , row1 column 1 , y less than x plus 4 , row2 column 1 , y greater than or equal to , x squared , plus 2 x plus 2 , end table
-
{
y
>
3
x
2
−
10
x
−
8
y
>
x
2
−
5
x
+
4
left brace . table with 2 rows and 1 column , row1 column 1 , y greater than 3 , x squared , minus 10 x minus 8 , row2 column 1 , y greater than , x squared , minus 5 x plus 4 , end table
