Chapter 1 Quadratic Equations in One Unknown

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1.1 Real Number System 1.3 Solving Quadratic Equations by Quadratic Conten ts 1 1.2 Solving Quadratic Equations by Factor Method Quadratic Equations in One Unknown Formula 1.4 Solving Quadratic Equations by Graphical Method 1.5 The Nature of the Roots of Quadratic Equations

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Chapter 1 Quadratic Equations in One Unknown

Transcript of Chapter 1 Quadratic Equations in One Unknown

Page 1: Chapter 1 Quadratic Equations in One Unknown

1.1 Real Number System

1.3 Solving Quadratic Equations by Quadratic

Contents1

1.2 Solving Quadratic Equations by Factor Method

Quadratic Equations in One Unknown

Formula

1.4 Solving Quadratic Equations by Graphical Method

1.5 The Nature of the Roots of Quadratic Equations

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A. Integers

1.1 Real Number System

The numbers 1, 2, 3, 4,…usually used in counting are called natural numbers. The natural numbers and their negatives, such as –1, –2, –3, –4,…together with the number 0 are called integers.

B. Rational Numbers

Any number that can be expressed in the form (where p and q are

integers and q ≠ 0) is called a rational number, for example,

qp

)10

7(7.0,)

1

4(4),

1

0(0,)

1

3(3,)

5

21(

5

14,

19

11,

2

5,

3

2

1

nFurthermore, any integer n can be written as , which is a fraction. So, all integers are also rational numbers.

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1.1 Real Number System

C. Characteristics of Rational Numbers in Decimals

A rational number that is not an integer can always be expressed as either a terminating decimal or a recurring (repeating) decimal.

Rational numbers whose decimals terminate

Rational numbers whose

decimals repeat

2.05

1 )1.0...(11111.0

9

1 i.e.,

25.04

1

375.08

3

)38.0...(83333.06

5 i.e.,

)742851.0...(571428571428.07

1 i.e.,

Table 1.1

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1.1 Real Number System

D. Irrational Numbers

E. Real Numbers

Real numbers are either rational numbers or irrational numbers.The line is called a real number line and each point on the line represents a real number.

.32 πand , example, for , an called is

integers two of ratio the as expressed be cannot that numberAny

number irrational

Fig. 1.1

line. nubmer real the on points different twoby drepresente

are number irrational the and number rational the 1.1, Fig. In 2411.

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1.2 Solving Quadratic Equations by Factor Method

A. Quadratic Equations

03413)1(4)1(.L.H.S 2

Now, if we put x = 1 into the equation (*), we have

and R.H.S. = 0

Therefore x = 1 satisfies equation (*).

Thus, 1 and 3 are called the roots of the equation (*).

Consider the equation

We call it a quadratic equation in one unknown because it is an equation of the second degree and it contains one unknown x.

).(........................................0342 xx

031293)3(4)3(L.H.S. 2

Similarly, putting x = 3 into the equation (*), we have

and R.H.S. = 0

Therefore x = 3 also satisfies equation (*).

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B. Using the Factor Method to Solve a Quadratic Equation

For two real numbers u and v, if uv = 0, then u = 0 or v = 0.

The above method of solving quadratic equations is called factor method.

If a quadratic equation can be factorized into a product of two linear factors, then we can solve it by using the following fact:

3or 103or 01

0342

xxx

xx< Apply the above fact,

.0342 xxSolve

Example:

1.2 Solving Quadratic Equations by Factor Method

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A. Completing the Square

A quadratic expression is called a perfect square if it can be factorized into a product of two identical linear factors.

The Method of Completing the Square

2

2

k

kxx 2By adding the term to , we have the following two

perfect squares.

22

2

22

kx

kkxx

22

2

22

kx

kkxxor

1.3 Solving Quadratic Equations by Quadratic Formula

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B. Using Quadratic Formula to Solve a Quadratic Equation

.2

4

)0( 0

2

2

a

acbbx

acbxax

by given are, equationquadratic the of roots The

Quadratic Formula

1.3 Solving Quadratic Equations by Quadratic Formula

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1.3 Solving Quadratic Equations by Quadratic Formula

a

cx

a

bx

a

cx

a

bx

acbxax

2

2

2

0

.0,0 where Solve

a

acb

a

bx

a

acb

a

bx

2

4

2

4

4

22

2

22

a

acbbx

2

42

Deriving the quadratic formula by completing the square

0 a

.expression side-left

the for square the Completing 22

2

22

a

b

a

c

a

bx

a

bx

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:222 xxy function the of graph the Consider

Fig. 1.2

In Fig. 1.2, the curve cuts the x-axis at the points x = –2.7 and x = 0.7. These are the values of x when y = 0, that is x2 + 2x – 2 = 0.

The roots of a quadratic equation ax2 + bx + c = 0 (a 0) can be obtained by finding the x-intercepts of the graph of y = ax2 + bx + c.

When there is no x-intercept, the quadratic equation has no real root.

1.4 Solving Quadratic Equations by Graphical Method

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.2

4

*).........(..........).........002

2

a

acbbx

acbxax

by given are (*) of roots the thatknow we1.3, Section In

( equationquadratic a Consider

.2

4

2

4

4),04(0

22

22

a

acbb

a

acbb

acbacb

and

has equation the So,

number. real positive a is then If 1: Case

:roots real unequal two

1.5 The Nature of the Roots of Quadratic Equations

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Content. has equation the So, number. real a not is then

,)(Δ If :3 Case

roots real noacb

acb

4

040

2

2

.2

4

*).........(..........).........002

2

a

acbbx

acbxax

by given are (*) of roots the thatknow we1.3, Section In

( equationquadratic a Consider

root. real double one has equation the Hence

. as simplified be can which, are equation the of roots the So,

zero. is then ,)(Δ If :2 Case

a

b

a

b

acbacb

22

0

4040 22

1.5 The Nature of the Roots of Quadratic Equations

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.4Δ

2

2

cbxaxyxacb

of graph the of intercepts- of number the determine to use also can We

.)intercepts- two are there (i.e., points. distinct two at axis- the cuts of graph the then , If : 1 Case

xxcbxaxy 20Δ

0 16

)3)(1(4)2(

32

2

2

that Note

1.19(a). Fig. in shown as Consider :Example xxy

Fig. 1.19(a)

.4 )0( 22 acbacbxaxy and Let

1.5 The Nature of the Roots of Quadratic Equations

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.intercept)-

oneonly is there (i.e.,only point one at axis-

the touches of graph the then ,Δ If : 2 Case

x

x

cbxaxy 20

).intercepts- no are there axis(i.e.,- the

cut not does of graph the then 0,Δ If :3 Case

xx

cbxaxy 2

Fig. 1.19(b)

Fig. 1.19(c)

that Note

1.19(b). Fig. in shown as Consider :Example

01616

)4)(1(4)4(

44

2

2

xxy

07

)2)(1(4)1(

2

2

2

that Note

).Fig.1.19(c in shown as Consider :Example

xxy

1.5 The Nature of the Roots of Quadratic Equations