quadraticequation
TRANSCRIPT
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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
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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
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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.
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2.1 Recognising Quadratic Equations
Exercise
Module Q.E page1
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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L1. Find the quadratic equation with roots -3
dan 5.
L2. Find the quadratic equation with roots 2
dan- 4.
Module page 9
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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
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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
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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)
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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
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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
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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]
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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.
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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
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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