8.3 1

Section 8.3 The Discriminant and the Nature of Solutions

The Discriminant Type and Number of Solutions Writing Equations from Solutions

8.3 2

Introducing … The Discriminant! is the Radicand Part of the Quadratic Equation

It predicts the types of solutions.If b2 – 4ac is positive: two different real numbers 0: one real (two equal real numbers) negative: two different complex numbers positive perfect square: two different rational numbers positive but imperfect: two different irrational numbers

a

acbbx

2

42

8.3 3

What Types of Solutions? b2 – 4ac

004129 2 xx

70852 xx

730372 2 xx

16042 x

144144)4)(9(4)12( 2

3225)8)(1(4)5( 2

2449)3)(2(4)7( 2

160)4)(1(4)0( 2

8.3 4

Writing Equations from Solutions We can use the reverse of the Principle of Zero Products (x – 2)(x + 3) = 0 means solutions x = 2 and x= -3 Think: x2 + x – 6 = 0 is equivalent to 2x2 + 2x – 12 = 0

Many quadratic equations can have the same solutions Find an equation having solutions:

x = 3 and x = 5/2 x = ±2i x = ± x = 0, x = -4 and x = 1

75

015112

0

0))(3(

2

215

2112

25

xx

xx

xx

04

04

0)2)(2(

2

22

x

ix

ixix

0175

0725

0)75)(75(

2

2

x

x

xx

0430)43(0)1)(4( 232 xxxxxxxxx

8.3 5

What Next? Quadratic Applications Section 8.4