EQUATIONS, INEQUALITIES AND ABSOLUTE VALUE

• QUADRATIC EXPRESSION AND EQUATION

• INEQUALITIES

• ABSOLUTE VALUES

QUADRATIC EXPRESSION AND EQUATION

Objectives:• Define quadratic expressions and equation• Solve quadratic equations by factorization,

completing the square methods and formula

QUADRATIC EXPRESSION AND EQUATION

Equation Quadratic a called is

0a and R c b,a, where02 cbxax

expression quadratic a called is

sign)equality an (without 2 cbxax

Eg :

5432 2 xx

432 2 xx

Solving Quadratic Equations

Quadratic equations can be solved by the following methods

a) when can be factorized

b) when cannot be factorized

02 cbxax

02 cbxax

a) when can be factorized02 cbxax

Example 1:

Solve the equation 062 2 xx

Solution:

Factorizing

2

3 x -2 x

03)(2xor 0 2)(x

0)32)(2(

062 2

xx

xx

The solution set is { -2 , 3/2 }

b) when cannot be factorized02 cbxax

•completing the square

•formula

completing the square

Solve the equation 0642 xx

Solution:

102

10 2

46)2(

2

46)2(

sidesboth to x)oft coefficien x (1/2 add 2

46

2

44

64

1 is xoft coefficien thesure make 064

2

22

222

2

22

x

x

x

x

xx

xx

xx

The solution set is {1.162, -5.162}

Test your power!!!!

Solve the equation 0132 2 xx

Answer:{1.781, -0.281}

Method using formula

a

acbbx

thenacbxaxIf

2

4

,0,0

2

2

0432equation theSolve 2 xx

Example 3

Solution:

351.2or 851.04

413

)2(2

)4)(2(433

0432

2

2

x

x

x

xx

The solution set is ????

Test your power again !!!!

052equation theSolve 2 xx

TYPES OF ROOTS OF A QUADRATIC EQUATION

Objectives:

a) Recognize the type of roots based on the discriminant

b) Relate the roots

c) Form a quadratic equations using identities

and

From the general equation, the types of the roots can be determined based on the value

of the discriminant,

,02 cbxax

:42 acb

i. If

ii. If

iii. If

roots realdistinct two,042 acbroots real equal two,042 acb

rootscomplex two,042 ab

Example

Determine the nature of the roots

1.

2.

3.

01682 xx

0642 xx

0593 2 xx

Example

roots. real equal twohas

,082 that if p of value theFind 2 pxx

Example

roots real have to

01equation for the

k of valuesofset theFind2 xkx

THE RELATIONSHIP BETWEEN THE ROOTS, and THE COEFFICIENTS OF A QUADRATIC EQUATION.

and

In general

0)(2 xx

0)(

or

0)(x

or

0roots) of(product roots) of sum(

2

2

2

aba

bx

x

xx

Important Identities:

})){((

))((

}3)){((

))((

2)(

233

22

233

22

222

Example

)

)

a)

of values thefind

,0162equation theof roots theare and If

33

22

2

c

b

xx

Solution

The equation has roots

Therefore,

,0162 2 xx and

2

1 and 3

82

12)3(

2)()

2

222

a

2

1-22

)2

18)(3(

))(() 2233

b

6

218

)22

c

To find a quadratic equation given the roots, the sum and product of the roots need to be found.

For example, the quadratic equation with roots 3 and 5 is

0158

0)53()53(

2

2

xx

isthat

xxx

Example

Given that are the roots of the quadratic equation

, find quadratic equation with roots

and

0123 2 xx

1

and 1

k. of value thefindother, the times twois

34equation theofroot one Given that 2 kxx

Again…test your power!!!!!!!!!!!

HOSTEL/WEEKEND JOBS

22

2

and are roots hoseequation w quadratic a b)

22 of value thea)

: find

,013equation theof roots theare and If

xx

INEQUALITIES

Objectives:

•Relate the properties of inequalities

•Define and solve linear inequalities

•Define quadratic inequalities and solve them using graphical method•Solve the quadratic by using analytical method:(i) Basic definition(ii) Real number line(iii) Table of signs.•Understand and solve rational inequalities involving linear andquadratic expressions.

LINEAR INEQUALITIES