The Chemistry Maths Book, Second Edition

2.5 Polynomials 43

Degree n 1

=

12 : quadratic function

f(x) 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

(2.16)

The quadratic function is usually written as

y 1 = 1 ax

2

1 + 1 bx 1 + 1 c (2.17)

A typical graph is shown in Figure 2.1 in Section 2.2. The shape of the curve is that of

a parabola. When the constant ais positive, the function has a single minimum value

(turning point), and is symmetrical about a vertical line that passes through the point

of minimum value. For the function in Figure 2.1,

f(x) 1 = 1 x

2

1 − 12 x 1 − 13 (2.18)

this minimum point has coordinates(x, y) 1 = 1 (1, −4). The graph crosses the x-axis,

whenf(x) 1 = 10 , at the two pointsx 1 = 1 − 1 andx 1 = 13. These are the roots of the quadratic

function, and they are the solutions of the quadratic equation

x

2

1 − 12 x 1 − 131 = 10 (2.19)

In this example the roots are easily obtained by factorization:

x

2

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

and the function is zero when either of the linear factors is zero:

(the symbol ⇒means ‘implies’)

Whilst it is possible to factorize a variety of quadratic functions by trial and error,

as in Examples 2.6, the roots can alwaysbe found by formula:

4

ax

2

1 + 1 bx 1 + 1 c 1 = 10

when

x (2.20)

bb ac

a

=

−± −

2

4

2

xx

xx

xx

2

230

10 1

30 3

−−=

+= ⇒ =−

−= ⇒ =





when

either

or







4

A clay tablet (YBC 6967, Yale Babylonian Collection) of the Old Babylonian Period (c. 1800–1600 BC)

has inscribed on it in the Sumerian cuneiform script the following problem (in modern notation): given

that xy 1 = 160 and x 1 − 1 y 1 = 17 , find xand y. The prescription given for the (positive) solution corresponds to

and. The method and prescriptive approach is almost

identical to that used by Al-Khwarizmi two and a half millennia later. Modern algebra became possible with the

development of a general abstract notation in the 15th to 17th centuries. One important step was taken by François

Viète (1540–1603). French lawyer, politician, cryptoanalyst, and amateur mathematician, he made contributions

to trigonometry and algebra. He is best remembered as the man who, in his In artem analyticem isagoge

(Introduction to the analytical art) of 1591, introduced the systematic use of symbols (letters) into the theory of

equations, distinguishing between constants and variables.

y=+−() ()72 60 72

2

x=++() ()72 60 72

2