2.4 To Form Quadratic Equations From Given Roots

2.1 Recognising Quadratic Equations

2.2 The ROOTs of a Quadratic Equation (Q.E)

2.3 To Solve Quadratic Equations2 0ax bx c

2.5 Relationship between and the roots of Q.E 2 4b ac

2.1 Recognising Quadratic Equations

Students will be taught to

1. Understand the concept quadratic equations and its roots.

Students will be able to:

1.1 Recognise quadratic equation and express it in

general form

QUADRATIC EQUATIONS

(i) The general form of a quadratic equation is ; a, b, c are constants and a ≠ 0.

2 0ax bx c

(ii) Characteristics of a quadratic equation:

(a) Involves only ONE variable,

(b) Has an equal sign “ = ” and can be expressed in the form ,2 0ax bx c

(c) The highest power of the variable is 2.

2.1 Recognising Quadratic Equations

Exercise

Module Q.E page1

Students will be taught to

2. Understand the concept of quadratic equations.

Students will be able to:

2.1 Determine the roots of a quadratic equation by

2.3 To Solve Quadratic Equations

( a ) Factorisation

( b ) completing the square

( c ) using the formula

Method 1 By Factorisation

This method can only be used if the quadratic expression can be factorised completely.

2 5 6 0x x Solve the quadratic equation

:Answer2 5 6 0x x

2 3 0x x

2 0 3 0x or x

2 3x or x

22 8 7 0x x Solve the quadratic equation by formula.Give your answer correct to 4 significant figures

:Answer

2 4

2

b b acx

a

Method 2 Formula2 4

2

b b acx

a

2( 8) ( 8) 4(2)(7)

2(2)x

a=2 , b =-8, c=7 8 8

4x

x = 2.707 atau 1.293

Method 3 By Completing The Square

- To express in the form of 2 0ax bx c 2a x p q

Solve by method of completing square2 4 5 0x x

2 4 5 0x x 2 2

2 45 0

22

44x x

2 42 5 0x

2 22 4 522 0x x

22 9 0x

22 9x

2 9x

2 3x

3 2x

1x

3 2x

5x

Simple Case : When a = 1

Method 3 By Completing The Square

- To express in the form of 2 0ax bx c 2a x p q

Solve by method of completing square2 3 2 0x x

2 22 3

2 03

2 23x x

23 9

2 42 0x

2 172 0

4x

2 172

4x

172

4x

172

4x

[a = 1, but involving fractions when completing the square]

2 3 2 0x x

x = - 0.5616

172

4x

x = 3.562 or

Method 3 By Completing Square

- To express in the form of 2 0ax bx c 2a x p q

Solve by method of completing square22 8 7 0x x

22 8 7 0x x

22

24 4

2 2

74 0

2x x

2 742 02

x

2 12 0

2x

If a ≠ 1 : Divide both sides by a first before you proceed with the process of‘completing the square’.

22 8 7 0

2 2 2 2

x x 2 first

2 74 0

2x x 1

22

x

2.707 or 1.293

Module Q.E page 4

2 ( 1) 6x x

1. Solve quadratic equation by factorisation.

2. Solve quadratic equation

by method of completing the square

3. By using formula,solve quadratic equation

2 4 5 0x x

2( 1) 1x

Students will be taught to

2. Understand the concept of quadratic equations.

Students will be able to:

2.2 Form a quadratic equation from given roots.

2.4 To Form Quadratic Equations from Given Roots

2.4 To Form Quadratic Equations from Given RootsIf the roots of a quadratic equation are α and β,

That is, x = α , x = β ; Then x – α = 0 or x – β = 0 ,

(x – α) ( x – β ) = 0

The quadratic equation is

2 ( ) 0x x

Sum of roots product of rootsx2x 0

Find the quadratic equation with roots 2 dan- 4.

x = 2 , x = - 4

2 ( 24)SOR

(2)( 4) 8POR

2 ( ) 0x x 2 ( ) 8 02x x 2 8 02x x

2 ( ) (Pr ) 0x sum of roots x oduct of roots

2.4 To Form Quadratic Equations from Given Roots

2 ( ) 0x x

2 ( )30

5

2 2x x

22 ( 1) 2 0x p x q Given that the roots of the quadratic equation

are -3 and ½ . Find the value of p and q.

13

5

2 2SOR

1( 3)( )

2

3

2POR

2 30

5

2 2x x

2 32 05x x

22 3 05x x and 2 ( 1) 22 0px x q

13,

2x x

1 5p

4p

2 3q

5q

Compare

L1. Find the quadratic equation with roots -3

dan 5.

L2. Find the quadratic equation with roots 2

dan- 4.

Module page 9

2 4 6 0,If and are the roots of theequation x x find the equation whose roots are

( ) 2 2a and 1:Step Find out SOR and POR of

2 4 6 0x x 1, 4 , 6a b c

SOR b

a

c

a

4

1

4

POR

6

:Step II Find out SOR and POR of2 2and

SOR 2 2 2( ) 2( )4

POR

(2 )(2 )

44( )624

:Step III Form equation2 ( ) ( ) 0x SOR x POR 2 ( ) ( ) 0x x

8

2 4 6 0,If and are the roots of theequation x x find the equation whose roots are

( ) 3 3a and 1:Step Find SOR and POR 2 4 6 0of x x 1, 4 , 6a b c

SOR b

a

c

a

4

1

4

POR

6

:Step II Find SOR and POR of 2 2and

SOR ( 3) ( 3)

6 64

POR

( 3)( 3)

3 3 9

3( ) 9 6 3( 94)

:Step III Form equation2 ( ) ( ) 0x SOR x POR 2 ( ) ( ) 0x x

2 3

2 2 3 0x x

Exercise 2.2.2 (Text book Page 34)

2 (a ) (b) (c ) (d)

3 ( a) (b ) ( c)

5.

10-3-2009

Skill Practice

2 (a ) (b) (c ) (d)

Students will be taught to

3. Understand and use the condition for quadratic equations to have

Students will be able to:

2.5.1 Relationship between and the roots of Q.E 2 4b ac

( a ) two different roots

( b ) two equal roots

( c ) no roots

3.1 Determine types of roots of quadratic equation from the value of .

2 4b ac

2.5 The Quadratic Equation 2 0ax bx c

2.5.1 Relationship between and the roots of Q.E 2 4b ac

Q.E. has two distinct/different /real roots.

The Graph y = f(x) cuts the x-axis at TWO distinct points.

2 4 0b ac 1Case

2.5 The Quadratic Equation 2 0ax bx c

2.5.1 Relationship between and the roots of Q.E 2 4b ac

Q.E. has real and equal roots.

2 4 0b ac 2Case

The graph y = f(x) touches the x-axis [ The x-axis is the tangent to the curve]

2.5 The Quadratic Equation 2 0ax bx c

2.5.1 Relationship between and the roots of Q.E 2 4b ac

Q.E. does not have real roots.

2 4 0b ac 3Case

Graph y = f(x) does not touch x-axis.

Graph is above the x-axis sincef(x) is always positive.

Graph is below the x-axis sincef(x) is always negative.

22 0x px q The roots of quadratic equation are -6 and 3

Find(a) p and q,(b) range of values of k such that does not have real roots.

22x px q k

( a) x = -6 and x=3

( x+6 )( x-3 )=0

2 3 18 0x x 2 36 062x x

Comparing

22x xp q k

P = 6 q = -36

2 32 66x x k

a = 2 b= 6 c=-36-k

2 62 036x kx

does not have real roots.2 4 0b ac 2 4(2)( ) 06 36 k 324 8 0k

40.5k

Module page 9

22x x k 1. Find the range of k if the quadratic equation has real and distinct roots.

2. Find the range of p if the quadratic equation has real roots.22 4 0x x p