The Chemistry Maths Book, Second Edition

44 Chapter 2Algebraic functions

The two roots are

(2.21)

and the quadratic has factorized form

ax

2

1 + 1 bx 1 + 1 c 1 = 1 a(x 1 − 1 x

1

)(x 1 − 1 x

2

) (2.22)

EXAMPLE 2.15The roots of the quadratic functionf(x) 1 =x

2

1 − 12 x 1 − 13.

We havea 1 = 11 ,b 1 = 1 − 2 ,andc 1 = 1 − 3 in formula (2.20). The roots are therefore

and the factorized form of the function isx

2

1 − 12 x 1 − 131 = 1 (x 1 + 1 1)(x 1 − 1 3).

EXAMPLE 2.16Find the roots of the quadratic functionf(x) 1 = 12 x

2

1 + 16 x 1 + 13.

We havea 1 = 12 ,b 1 = 16 ,andc 1 = 13 in formula (2.20), and the roots are

0 Exercises 44–46

The quantity

b

2

1 − 14 ac (2.23)

in (2.20) is called the discriminantof the quadratic function. Its value in Examples

2.15 and 2.16 is positive, and the function has two real roots, but in other examples it

can have zero or negative value. A graphical explanation of the three possible types of

discriminant is shown in Figure 2.8.

x=

−± −

=−±

()

63624

4

1

2

33

x=

+± +

=± =−

2412

2

12 1 3or

x

bb ac

a

x

bb ac

a

1

2

4

2

4

2

=

−+ −

,=

−− −

y

x

x 1

x 2

•

b

2

− 4 ac> 0

2 differentrealroots

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Figure 2.8