2.03 add math module 03 quadratic equations · pdf file1.12.2008 · spm quadratic...
Post on 31-Jan-2018
232 Views
Preview:
TRANSCRIPT
QUADRATIC
ADDITIONAL MMOD
http://mathsmoz
http://sahatmozac.blogspot.com
ATHEMATICS
EQUATIONS
ULE 3
ac.blogspot.com
1
CHAPTER 2 : QUADRATIC EQUATIONS
MODUL 3
2.1 CONCEPT MAP 2
2.2 GENERAL FORM 3
2.2.1 Identifying 3Exercises 1 3
2.2.2 Recognising general form of quadratic equation 4.ax2 + bx – c = 0Exercises 2 4
2.3 SOLVING QUADRATIC EQUATIONS 6
2.3.1 Factorisation 6Exercises 1 6
2.3.2 Completing the square 8Exercises 1 8
2.3.3 Quadratic formula 10Exercises 1 10
2.4 PASS YEARS QUESTIONS 12
2.5 ASSESSMENT 13
ANSWERS 15
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
CHAPTER 2 : QUADRATIC EQUATIONS
MODUL 3
2.1 CONCEPT MAP
QUADRATIC EQUATIONS
x x = 0
xx 2 = 0
> = 0 (Positive)
Two differentroots
http://m
http://sahatmozac.blogspot.com
GENERAL FORM
2
ROOTSx = ,
.ax2 + bx + c = 0
b2 – 4ac
roots
< = 0 (n
Noro
Types of roots
athsmozac.blogspot.co
Factorization
Completing the square
Formula
egative)
realots
= 0
TwoEqual
m
3
2.2 GENERAL FORM
2.2.1 Identifying
Example 1
4x + 3 = 2x2x + 3 = 0
The highest power of variable x is 1Therefore 4x + 3 = 2x is not a quadraticequation
Example 2
x(x + 5) = 7x2 + 5x - 7 = 0
The highest power of variable x is 2Therefore x(x + 5) = 7 is a quadraticequation
Exercises 1Identify which of the following are quadratic equation
1. 3 =x2
5.2. x(2x + 3) = x - 7
3. ( x + 4 )(2x – 6) + 3 = 0 4. (3m + 5)2 = 8m
5. x (7 - 2x + 3x2) = 0 6. 3x2 – 5 = 2x( x + 4)
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
4
2.2.2 Recognising general form of quadratic equation .ax2 + bx – c = 0
Example 1
.x2 = 5x – 9
.x2 – 5x + 9 = 0
Compare with the general form.ax2 + bx – c = 0
Thus, a = 1, b = -5 and c = 9
Example 1
4x =x
xx 22
4x(x) = x2 – 2x4x2 - x2 – 2x = 03x2 – 2x = 0
Compare with the general formThus, a = 3, b = - 2 and c = 0
Exercises 2Express the following equation in general form and state the values of a, b and c
1. 3x =x2
5
.2. (2x + 5) =x
7
3. x( x + 4 ) = 3 .4. (x – 1)(x + 2) = 3
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
5
5.x
4=
x
x
5
3 6. x2 + px = 2x - 6
7. px (2 – x) = x – 4m 8. (2x – 1)(x + 4) = k(x – 1) + 3
9. (7 – 2x + 3x2) =3
1x10. 7x – 1 =
x
xx 22
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
6
2.3. SOLVING QUADRATIC EQUATIONS
2.3.1 Factorisation
Exercises 3Solve the following quadratic equation by factorisation
1. x2 + 3x - 4 = 0 2. x2 -2x = 15
3. 4x2 + 4x – 3 = 0 4. 3x2 - 7x + 2 = 0
Example 1
.x2 + 6x + 5 = 0 x + 3 3x( x + 3)(x + 2) = 0 .x + 2 2x.x + 3 = 0 or x + 2 = 0
.x = -3 x = - 2 .x2 + 6 5x
Therefore, The roots of the equation are.x = -3 and -2
Example 2
4(x +3) = x(2x – 1)4x + 12 = 2x2 - x2x2 - 5x - 12 = 0(2x + 3)(x - 4) = 02x + 3 = 0 or x - 4 = 0
.x =2
3x = 4
Therefore, The roots of the equationare
.x =2
3and 4
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
7
5. x2 = 3x – 2 6. x(2x - 5) = 12
7. 8x2 + x = 21(1 – x) 8. (2y – 1)(y + 4) = -7
9. 4y -y
1= 3 10.
23
67
m
m= m
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
8
2.3.2. Completing the square
Example 1
.x2 – 6x + 7 = 0.x2 - 6x = -7
.x2 – 6x +2
2
6
= - 7 +
2
2
6
.x2 – 6x + (-3)2 = -7 + (-3)2
(x - 3)2 = 2
.x – 3 = 2
.x = 3 2
.x = 3 + 2 or 3 - 2
.x = 4.414 or 1.586
Example 2
2x2 -5x – 1 = 02x2 – 5x = 1
.x2 -2
5x =
2
1
.x2 -2
5x +
2
4
5
=
2
1+
2
4
5
x =
2
1+
16
25
=16
33
.x -4
5=
16
33 =
4
33
.x =4
5+
4
33or
4
5-
4
3
.x =4
335 or
4
335
.x = 2.686 or -0.186
Exercises 4Solve the following quadratic equation by completing the square
1. (x + 3 )2 = 16 2. (5x - 4)2 = 24
Rearrange in theform.x2 + px = q
Add
2
2
...
xoftcoefficien
To both sides
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
Change thecoefficient
2
2
4
5
3
of x to 1
9
3. x2 - 8x + 12 = 0 4. 3x2 + 6x – 2 = 0
5. 5x2 – 7x + 1 = 0 6. 2x2 – 3x – 4 = 0
7. (x + 1)(x - 5) = 48. 1 -
x
1=
22
3
x
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
10
Quadratic formula
Example 1
.x2 + 5x + 2 = 0.a = 1, b = 5, c = 2
Using the formula x =a
acbb
2
42
.x =
12
21455 2
.x =2
8255 =
2
175
.x =2
175 or
2
175
= - 0.438 or - 4.562
Example 2
3x2 = 4x + 23x2 - 4x – 2 = 0.a = 3, b = -4 , c = - 2
Using the formula
.x =
32
23444
=
6
24164
=6
404
.x =6
404 or
6
404
= 1.721 and – 0.387
Exercises 5Solve the following quadratic equations by using the quadratic formula
1. x2 – 11x + 28 = 0 2. –x2 – 3x + 5 = 0
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
11
3. 2x2 + 11x + 9 = 0 4. 3x2 + 14x – 9 = 0
5. 10x(2x – 1) – 8 = x(2x + 35) 6. (x – 1)(4x – 9) + 7 = 10x
7.3
211
v
v= 2v 8.
1
132
2
xx
xx= 2
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
12
9.5
23 x+
2
3
x
x= 1 10.
x
x1+ 3 =
5
7
x
2.4 PAST YEARS QUESTIONS
SPM 2001. PAPER 1 Question 3
1. Solve the quadratic equation 2x(x + 3) = ( x + 4)(1 - x). Give your answercorrect to four singnificant figures.
SPM 2003. PAPER 1 Question 3
1. Solve the quadratic equasion 2x(x – 4) = (1 – x)(x + 2)Give your answer correct to four significant figures.
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
13
2.5 ASSESSMENT ( 30 minutes)
1. Express 3x2 – 2px = 5x - 7p in genegal from
2. Find the roots of the equation 2x2 + 5x = 12
3. Find the roots of2
1
x=
3
x,
4. By using the quadratic formula, solve the equation 2x2 – 5x – 1 = x(4x - 2)
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
14
5. Solve the quadratic equation (5x – 3)(x + 1) = x(2x – 5) .Give your answer correct to four significant figures.
6. Given the equation x2 + 4x – 5 = (x – a)2 + b , find the values of a and b
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
15
ANSWERS
Exercises 11. No 2. Yes3. Yes 4. Yes5. No 6. Yes
Exercises 21. 6x2 – 5 = 0 a= 6, b = 0, c = -52. 2x2 + 5x – 7 = 0 a = 2 , b = 5 , c = -73. x2 + 4x – 3 = 0 a = 1, b = 4 , c = - 34. x2 + x – 5 = 0 a = 1, b = 1, c = -55. x2 + 7x – 20 = 0 a = 1, b = 7, c = -206. x2 + (p – 2)x + 6 = 0 a= 1, b=(p – 2), c= 67. px2 + (1 – 2p)x – 4m = 0 a = p , b = (1 – 2p) , c = -4m8. 2x2 + (7 - k) x + k – 7 = 0 a = 2, b = (7 - k), c = (k – 7)9. 9x2 - 7x + 20 = 0 a = 9, b = -7 c = 2010. 6x2 + x = 0 a = 6, b = 1, c = 0
Exercises 31. 1, -4 2. 5, -3
3.2
1, -
2
34.
3
1, 2
5. 1, 2 6. 4,2
3
7.2
7,
4
37.
2
1, - 3
9.4
1, 1 10. 1,
3
7
Exercises 41. 1, -7 2. 1.77, - 0.173. 2, 6 4. 0.457 , - 1.4375. 0.161, 1.239 6. 2.351 , - 0.8517. 5.60, -1.60 8. 1.823, -0.823
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
16
Exercises 51. 4, 7 2. 1.191, - 4.191
3. -1,2
94. 0.573 , -5.239
5. 2.667, - 0.169 6. 0.810, 4.94
7. 2,2
18. 5.192, -1.925
9. 2.812, - 0.119 10. -0.403, -3.069
PAST YEARS QUESTIONS
1. 0.393, -3.393 2. 2.591, -0.2573
ASSESSMENT
1. 3x2 – (2p + 5)x + 7p = 0
2.2
3, 4
3. – 3, 14. - 0.5, -15. 0.370, -2.706. a = 2, b = -9
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
17
SPM
QUADRATIC EQUATIONS
ADDITIONAL MATHEMATICSMODULE 4
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
18
CHAPTER 2 : QUADRATIC EQUATIONS
MODUL 4
2.1 CONCEPT MAP 2
2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS and 3
2.2.1 Form a quadratic equation with the roots 4
Exercises 1 4
2.2.2 Determine the sum of the roots and product of the roots of the 5following quadratic equations.Exercises 2 5
2.3. TYPES OF ROOTS QUADRATIC EQUATION 6
2.3.1 Determine the types of roots for each of the following 6quadratic equationsExercises 3
2.4 SOLVING PROBLEMS INVOLVING (.b2 - 4ac ) 7
2.4.1 Find the values of k for each of the following quadratic 7equations which has two equal roots
Exercises 4 7
2.4.2 Find the range of values of h for each of the following 8quadratic equations which roots are different
Exercises 5 8
2.4.3 Find the range of values of m for each of the following 9quadratic equations which has no roots
Exercises 6 9
2.5 PASS YEARS QUESTIONS 10
2.6 ASSESSMENT (30 MINUTES) 12
ANSWERS 14
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
CHAPTER 2 : QUADRATIC EQUATIONS
MODUL 4
2.1 CONCEPT MAP
S
QUADRATIC EQUATIONS
.x =
GENERAL FORM
Sum
Pro
TY
http://mathsm
http://sahatmozac.blogspot.com
ROOTS
and x =The quadratic equation
FORMING A QUADRATIC EQUATIONFROM GIVEN ROOTS
of the roots
duct of the roots
The quadratic equation
19
PES OF ROOTS
ozac.blogspot.com
20
2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS and
Notes
If x1 = a and x2 = bThen (x – a) = 0 and (x – b) = 0
(x – a)(x – b) = 0.x2 – bx – ax + ab = 0.x2 – (a + b)x + ab = 0
The quadratic equation with roots and is written as
.x2 – ( )x + = 0 …….(1)
From general form ax2 + bx + c = 0
a
ax 2
+a
bx+
a
c= 0
.x2 +a
bx+
a
c= 0 ……………
Compare with the equations (1) and (2) .x2 – (
x2 +a
bx+
– ( ) =c
b
( ) = -c
b( The sum of the root
=a
c( The product of the
Sumof theroots
Product.of theroots
Product.of the
http://mathsmozac.blogsp
http://sahatmozac.blogspot.com
Sumof the
………(2)
)x + = 0 ……….….(1)
a
c= 0 ……………………(2)
s)
roots)
roots roots
ot.com
21
2.2.1 Form a quadratic equation with the roots
Exercises 1Roots and
Sum of.the roots( )
Product of.the roots
The puadratic equation
.x2 – ( )x + = 0
Example 13 , 2 5 6 x2 – (5)x + 6 = 0
Example 2
4
1, - 3
4
1+ (-3)
=4
121=
4
11
4
1(- 3) =
4
3.x2 –
4
11x +
4
3= 0
4x2 + 11x – 3 = 0
a) 4 , -7
b)
2 ,3
1
c)
3
1,
2
1
d)
5
1,
3
2
e)
3k,5
6k
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
22
2.2.2 Determine the sum of the roots and product of the roots of the followingquadratic equations.
Exercises 2The puadratic equation Sum of
.the rootsProduct of.the roots
Example 1.x2 – 6x + 9 = 0 6 9
Example 1.9x2 + 36x - 27 = 0
9
9 2x+
9
36x-
9
27= 0
.x2 + 4x – 3 = 0
-(4) = -4 -3
a) .x2 + 73x - 61 = 0
b) 7x2 - 14x - 35 = 0
c) 2x(x + 3) = 4x + 7
d) 2x +x
2=
4
1
e)4x2 + kx + k – 1 = 0
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
2.3. TYPES OF ROOTS QUADRATIC EQUATION
2.3.2 Determine the ty
Exercises 31. 2x2 - 8x + 3 = 0
3. 3x2 = 7x - 5
Example 1
a) . x2 – 12x + 27 = 0. a = 1, b = -12 , c = 27
.b2 - 4ac = (-12)2 – 4(1)(27= 144 – 108= 36 > 0
Thus, x2 – 12x + 27 = 0Has two different roots
. b2 - 4ac > 0
. Two different
ht
http://sahatmozac.blogspot.com
pes o
)
E
b.
.b
.x =
a
acbb
2
42
.b2 - 4ac
. b2 - 4ac < 0
tp://
. b2 - 4ac = 0
23
f roots for each of the following quadratic equations
2. -2y2 + 6x + 3 = 0
4. 4x2 = a(4x - a)
xample 2
) .4 x2 – 12x + 9 = 0a = 4, b = -12 , c = 9
2 - 4ac = (-12)2 – 4(4)(9)= 144 - 144= 0
Thus, 4x2 – 12x + 9 = 0Has two equal roots
Example 3
c) .2 x2 – 7x + 10 = 0. a = 2, b = -7 , c = 10
.b2 - 4ac = (-7)2 – 4(2)(10)= 49 – 80= - 31< 0
Thus, 2x2 – 7x + 10 = 0Has no real roots
. Two equal root . No real roots
mathsmozac.blogspot.com
24
2.4 SOLVING PROBLEMS INVOLVING (.b2 - 4ac )
The discriminant b2 - 4ac of the quadratic equation can be used toa) find an unknown value in an equation
Example 1
The quadratic equation x2 - 2px + 25 = 0Has two equal roots. Find the value p
x2 - 2px + 25 = 0Thus, a = 1, b = -2p , c = 25Using .b2 - 4ac = 0
(-2p)2 – 4(1)(25) = 04p2 - 100 = 0
4p2 = 100p2 = 25
p = 25
p = 5
Example 2
The quadratic equation x2 – 2kx = -(k – 1)2
Has no roots. Find the range of values of k
x2 – 2kx = -(k – 2)2
x2 – 2kx + (k – 2)2 = 0Compare with ax2 + bx + c = 0Thus .a = 1, b = - 2k , c = (k – 2)2
Using b2 - 4ac < 0(-2k)2 – 4(k2 -2k + 1)< 04k2 - 4k2 + 8k – 4 < 0
8k – 4 < 08k < 4.k < 4
2.4.1 Find the values of m for each of the following quadratic equations whichhas two equal roots
Exercises 4
1. mx2 - 4x + 1 = 0 2. x2 – 6x + m = 0
3. x2 – 2mx + 2m + 3 = 0 4. x2 - 2mx - 4x + 1 = 0
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
25
5. x2 + 6x - 9 = m (2x - 3) 6. x2 + 2(x + 2) = m(x2 + 4)
2.4.2 Find the range of values of h for each of the following quadratic equationswhich roots are different
Exercises 5
1. x2 - 6x - h = 0 2. hx2 – 4x – 3 = 0
3. x2 + 6x + h + 3 = 0 8. 2hx2 + 4x + 1 = 0
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
26
5. x(5 – 2x) = h + 2 6. 2(hx2 – 1) = x(x – 6)
2.4.3 Find the range of values of m for each of the following quadratic equationswhich has no roots
Exercises 6
1. 2x2 + 2x - m = 0 2. mx2 + 3x - 3 = 0
3. x2 + 2x + m - 3 = 0 4. 3x2 + 1 = 2(m + 3x)
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
27
5. (h + 1)x2 + 8x + 6 = 0 6. (m - 3)x2 + 2(1 – m)x = -(m + 1)
2.5 PASS YEARS QUESTIONS
SPM 2001/P1 Question 4
1. Given that -1 and h are roots of the quadratic equation(3x – 1)(x – 2) = p(x – 1), where p is a constant, find the values of h and p
SPM 2002/P1 Question 4
2. Given that 3 and n are roots the equation (2x + 1)(x – 4) = a(x – 2),where a and n are constants, find the values of a and n.
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
28
SPM 2002/P1 Question 12
3. It is given that3
and
3
are roots of the quadratic equation px(x – 1) = 3q + x
If = 12 and = 3, find the values of p and q
SPM 2004/P1 Question 4
4. Form a quadratic equation which has the roots – 5 and4
3,
Give your answer in the form of ax2 + bx + c = 0 ,Where a, b, and c are constans
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
29
2.6 ASSESSMENT (30 MINUTES)
1. Form a quadratic equation which has the roots 3 and - 4
2. Given that -3 and 4 are roots of the quadratic equation x2 + ax = bFind the values of a and b
3. The quadratic equation x2 - kx + 2k = 4 has roots 2 and 6Find the values of k
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
30
.4 Find the values of h if the equation x2 = 4hx - 36 has equal roots
5. The quadratic equation kx2 - 2(3 + k)x = 1 – k has no real roots.Find the range of vales of k
6. The quadratic equasion x(x – 2m) = - ( 3m + 4) has equal roots, finda) the value of mb) the roots
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
31
ANSWERS
Exercises 1
a. -3, -28, x2 + 3x – 28 = 0 b.3
5,
3
2, , 3x2 -5x – 2 = 0
c.6
5,
6
1, 6x2 - 5x + 1 = 0 d.
15
7,
15
2, 15x2 + 7x – 2 = 0
e.5
21k,
5
18 2k, 5x2 - 21kx – 18 k2 = 0
Exercises 2
a. -73 , - 61 b. 2, - 5
c. -1,2
7, d.
8
1, 1
e.4
k,
4
1k
Exercises 3
1. Two different roots 2. Two different roots3. No real roots 4. Two equal roots
Exercises 4
1. m = 4 2. m = 93. m = 3 and -1 4. m = - 3 and -1
5. m = 3 and 6 6. m =2
3and
2
1
Exercises 5
1. h > - 9 2. h >3
4
3. h < 6 4. h < 2
5. h <8
96. h >
4
7
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
32
Exercises 5
1. m <2
12. m <
4
3
3. m > 4 4. m < -3
5. m >3
56. m >
3
1
PASS YEARS QUESTIONS
1. P = - 6 , h =3
4
2. a = -7 n = 3
3. p =3
1, q =
27
1
4. 4x2 + 17x – 15 = 0
ASSESSMENT
1. x2 + x - 12 = 02. a = -1, b = 123. k = 84. h = 3
5. k <7
9
6. m = 4, -1x = 4, -1
http://mathsmozac.blogspot.com
http://sahatmozac.blogspot.com
top related