574 CHAPTER 9 Quadratic Equations
OBJECTIVES Some quadratic equations have no real number solutions. For example, solving the
equation from Example 6of Section 9.2, by the quadratic for-
mula leads to the values
These are not real numbers,
since the radicand is.To ensure that every quadratic equation has a solution, we need a new set of num-
bers that includes the real numbers. This new set of numbers is defined with a new
number i,called the imaginary unit,such that
and thus,OBJECTIVE 1 Write complex numbers as multiples of i.We can write num-
bers such as and as multiples of i, using the properties of ito
define any square root of a negative number as follows.2 - 4 , 2 - 5 , 2 - 8
i 2 1 , i^2 1.- 16
- 8 2 - 16
8
.
4 p^2 + 8 p+ 5 =0,Complex Numbers
9.4
1 Write complex
numbers as
multiples of i.
2 Add and subtract
complex numbers.
3 Multiply complex
numbers.
4 Divide complex
numbers.
5 Solve quadratic
equations with
complex number
solutions.
- 42.x^2 -
100
81
x^2 - = 04
15
=-4
15
x1
2
n^2 - n=15
2
5 x^2 + 19 x= 2 x+ 12z^2 -5
12
z=1
6- 3 x^2 + 4 x=- 4
x^2 - x+ 3 = 0 4 m^2 - 11 m+ 8 =- 29 x^2 = 1613 x+ 42 t 115 t+ 582 =- 481
5
x^2 +x+ 1 = 0r^2
2
+7 r
4
+11
8
= 02 p^2 = 2 p+ 1 3 m 13 m+ 42 = 7 5 x- 1 + 4 x^2 = 01 x+ 221 x+ 12 = 10 16 x^2 + 40 x+ 25 = 0 4 x^2 =- 1 + 5 x0.3x^2 +0.5x=-0.1 5 z^2 - 22 z=- 8 z 1 z+ 62 + 4 = 08 z^2 = 15 + 2 z 3 x^2 = 3 - 8 x 0.1x^2 - 0.2x=0.12 t^2 + 1 =t - 2 x^2 =- 3 x- 2 - 2 x^2 +x=- 113 r- 722 = 24 17 p- 122 = 32 15 x- 822 =- 612 s- 122 = 10 1 x+ 622 = 121 15 x+ 122 = 36z 1 z- 92 =- 20 x^2 + 3 x- 2 = 0 13 x- 222 = 9For any positive real number b, 2 bi 2 b.2 bhttp://www.ebook777.com
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