SOLUTION OF QUADRATIC EQUATIONS BY FACTORISATION
SOLVE THE FOLLOWING EQUATION:
1.1 x²-3x-10=0
(x-5)(x+2)
x=5 x=-2
ARE YOU SURE YOU UNDERSTAND WHY (x-5)(x+2)?
SOLVE FOR X:
1.2 x(x-1)=6
x²-x=6
x²-x-6=0
(x-3)(x+2)=0
x=3 x=2
DO THE FOLLOWING EXERCISES:
1.3 (x-3)(x-2)=12
1.4 x(x+4)=21
1.5 x(x-1)=4(3x-10)
1.6 x²+2x-3=12
1.7 x(x-16)=3(24-5x)
1.8 (2x-5)(3x+2)=2(3x-11)
1.9 (2x+1)(x-5)-(x-3)²+(x-2)(x+5)=-23-x
1.10 4(x-1)(x+1)-4(x-2)=12(x-1)+1
ANSWERS FOR QUESTION 1:
SOLUTION OF QUADRATIC EQUATIONS WITH FRACTIONS
2.1 SOLVE:
x+3 1 1 7
______ + ______ = ______ - ______
x²-4 x²+x-2 2-x 4-x²
x+3 1 1 7
____________ + ____________ = _____ - ___________
(x-2)(x+2) (x+2)(x-1) 2-x (2-x)(2+x)
LCD=(x-2)(x+2)(x-1)
Now multiply both sides of the equation by the LCD:
(x+3)(x-1)+1(x-2)=-1(x+2)(x-1)+7(x-1)
x²+2x-3+x-2=-x²-x+2+7x-7
2x²-3x=0
x(2x-3)=0
x=0 or x= 3
___
2
2.2 x-3 5 4
______ - _______ = _______
x²+3x+2 x²-4 -x-1
x-3 5 -4
________ - ________ = ________
(x+2)(x+1) (x-2)(x+2) x+1
Multiply both sides of the equation by the LCD:
(x-3)(x+2)-5(x+1)=-4(x-2)(x+2)
x²-5x+6-5x-5=-4x²+16
5x²-10x-15=0
x²-2x-3=0
(x-3)(x+1)=0
x=3 x=-1
DO THE FOLLOWING EXERCISES:
2.3 x= 4
____
x
2.4 x-2= 8
___
x
x 6
2.5 ____ = _____
x-2 x-1
x+2 3 1
2.6 _____ - _____ = ______
x+1 x-2 x+1
a+1 -2 a+2
2.7 ____ = _____ + ______
a-1 a+2 1-a
2.8 6 x+2 x+3 2x-2
1 + ___ + _____ = ______ + _______
x+1 x+1 x-1 1-x²
2.9 2 2 1 3
_______ = ______ + _________ + ________
x²+3x+2 1-x² x²+3x+2 x²+x-2
2.10 3 x+3
_____ + x + 5 = ______
x x
ANSWERS FOR QUESTION 2:
SOLUTION OF QUADRATIC EQUATIONS BY FACTORISATION
USING A SUITABLE A SUITABLE SUBSTITUTION
SOLVE FOR x: x²-2x=18- 45
_____
x²-2x
Let x²-2x=K
Then: K=18K-45
K²=18K-45
(K-15)(K-3)=0
K=15 or K=3
x²-2x=15 or x²-2x=3
x²-2x-15 or x²-2x-3
(x+3)(x-5)=0 or (x-3)(x+1)=0
x=-3 or x=5 or x=3 or x=-1
SOLVE THE FOLLOWING EQUATIONS
3.1 x4-13x+36=0
3.2 1 2
____ - _____ - 3 = 0
x² x
3.3 6y-2+y-1-2=0
3.4 (x²-3x)²-20=8(x²-3x)
3.5 (x²+3x)²-2(x²+3x)-8=0
3.6 y²-y-3= 9
__________
y²-y-3
3.7 (x²-2x)²-2(x²-2x)-3=0
3.8 (x²-5x)²=36
3.9 (2x²+5x)²-10x²-25x-14=0
3.10 x²-5x+2- 4
________ =0
x²-5x+2
ANSWERS FOR QUESTION 3:
SOLUTION OF QUADRATIC EQUATIONS BY SQUARING BOTH SIDES
4.1 SOLVE FOR x: SQR(x+6)=x
x+6=x²
x²-x-6
(x-3)(x+2)
x=3 x=-2
DO THE FOLLOWING EXERCISES:
4.2 SQR(5x+6)=x
4.3 SQR(5x-25)-SQR(x-1)=0
4.4 SQR(x²+5x+11)-2x=1
4.5 x-4=3SQR(x-6)
4.6 2SQR(2- X +4=X
___
2
4.7 SQR(2x+3)=x+2
4.8 SQR(x+5)=SQRx+1
4.9 2x+SQR(8x-3)=0
4.10 SQR(2-7x)+2=x
ARITHMETIC AND GEOMETRIC PROGRESSIONS
1. Addition of a regular amount-called an
Arithmetic Progression
2. Multiplication by a regular amount-called
a Geometric Progression
Example of Arithmetic Progression
2;5;8;11.....
The amount added to each is 3
SECOND TERM MINUS FIRST
5-2=3
THIRD TERM MINUS SECOND
8-5=3
THE Common DIffernce (d) is 3
TERM Tn=a+(n-1)d
SUM Sn=n/2[2a+(n-1)d]
MEAN A.M.= (a+b)
_____
2
Example:
A:Find the 16th term of the sequence 4;7;10...
7-4=3
10-7=3
Tn=a+(n-1)d
Tn=4+(16-1)3
=4+(15)3
=4+45
=49
B:Find the A.P. of which the 7th term
is 10 and the 13th is -2
Tn=a+(n-1)d
10=a+(7-1)d
10=a+6d
Tn=a+(n-1)d
-2=a+(13-1)d
-2=a+12d
10=a+6d
-2=a+12d
_________
12=-6d
12=d
___
-6
-2=d
10=a+6d
10=a+6(-2)
10=a-12
a=22
So progression is 22;20;18;16;14;.......
EXAMPLE of GEOMETRIC PROGRESSION(G.P.)
2;8;32
The amount each term is multiplied by to get the
next is 4....and
this is found from the DIVISION TEST
DIVIDE second by the first
8
____ = 4
2
DIVIDE third by the second
32
____ = 4
8
The answer must be the same for Division Test
to be passed
This answer is called the Common Ratio
and represented by 'r'.
G.P.
TERM Tn=arn-1
SUM= Sn=a(1-rn)
__________________
1-r
G.M.= + SQR ab
-
S= a
____
1-r
A:FIND THE 7th of the progression 16;8;4;......
8-16=-8
4-8=-4
Subraction test fails
8 1
___ = ___
16 2
4 1
___ = ___
8 2
Tn=ar(n-1)
Tn=(16)(1)(7-1)
_
2
=(16)( 1 )6
__
2
= 24
_____
26
= 1
__
22
= 1
___
4
LOGARITHMS
log28=x
the expression could also be written as:
2x=8
LogLaws
1. When numbers are multiplied their logs will
be added log(2x3)=log2+log3
2. When numbers are divided their logs will
be subtracted
log 4 = log4-log6
___
6
3. The exponent of a term becomes the coeficient
of its log term
logx3=3logx
4 Change of base law
logwy
_______ =logxy
logwx
EXPONENTIAL LAWS
1.When like bases are multiplied
their exponents are added
x3.x2=x3+2=x5
2.When like bases are divided
their exponents are subtracted.
x4
___ = x4-6=x-2
x6
3.Power to Power Laws
(x3)4
=x12
simplify log3216
= log16
______
log32
= log24
_______
log25
= 4log2
______
5log2
= 4
___
5
simplify log354-log32
log354
_____
2
=log327
log 27
= _______
log 3
= log 33
_______
log 3
= 3log3
______
log3
= 3
simplify: log28+log82+log31
log28+log82+0
log8 log2
_____ + ______
log2 log23
log23 log2
= _____ + ______
log2 log23
3log2 log2
= _____ + ______
log2 3log2
= 3+ 1
__
3
9+1
= ____
3
10
= _____
3
CALCULUS
The calculus is the most powerful mathematical
invention of modern times.The credit for its
discovery has been for both Sir Isaac Newton and
Leibnitz,the great German mathematician.A branch
of mathematics that is concerned with the study
of rates of change(differntial calculus)and the
areas and volumes of figures with curved
bounderies(integral calculus).
1. If f(x)=x²
then f'(x)=2.1x2-1
f'=symbol for the first differential
[(exponent)(coeficient)
x(variable power-1)]
2. If y=3x²=2x-2
y'=(2)(3)x2-1
y'=6x-4x-3
3. If f(x)=16x²-32x²
Dx=
(2)(16)x2-1-(3)(32)x3-1
Dx=32x-96x²
Uses of the differential
1.The first and most obvious use of the differential
is to find the gradient at a given point on a curve.
e.g Find the gradient when x=2 on the curve
f(x)=x3-4x²
To find gradient formula f'(x)=3x²-8x
when x=2 f'(x)=3(2)²-8(2)
=12-16
=-4
Gradient is therefore -4 at this point.
2.This now enables us to find the equation of either
the tangent or the normal drawn at this point. e.g
Find the equation of the tangent drawn to the curve
y=2-4x²+x3at the point where x=1
To find gradient formula y'=-8x+3x²
Gradient when x=1: y'=-8(1)+3(1)²
=-8+3
= -5
Gradient of tangent=-5
Equation of tangent y=mx+c
or y=-5x+c
But tangent and curve have a common point-where x=1
To find y value at this point y=x3-4x²+2
y=(1)3-4(1)²+2
=1-4+2
=-1
To find 'c' in the tangent equation y=mx+c
y=-5x+c
At(1;-1) -1=-5(1)+c
-1=-5+c
4=c
So the tangent equation is y= -5x+4
Here is an example of finding the first derivative
by first principles
f(x)=(3x-5)
limf(x+h)-f(x) = lim(3(x+h)-5)-(3x-5)
h->0___________ h->0_________________
h h
= lim 3x+3h-5-3x+5
____________
h->0 h
= lim 3h
h->0 _____
h
= lim 3
h->0
=3
MATRIX ALGEBRA(ADVANCED TERTIARY MATHEMATICS)
What is a Matrix?
Whenever one is dealing with data,there should
be concern for organizing them in such a way that
they are meaningful and can be readily identified.
Summarizing data in tabular form serves this function.
Income tax tables are an example of this type
of organisation.A matrix is a common device
for summarizing and displaying numbers or data.
A= {1 3} B= {-3 2}
{4 -2} { 0 4}
A+B= {1 3} + {-3 2}
{4 -2} { 0 4}
= {1+(-3) 3+2}
{4+0 -2+4}
find the matrix cofactors for (2x2) matrix
A= {5 -4} (-2)
{2 -2} Submatrix
a'11=(-1)1+1(-2)
=(-1)²(-2)
=(1)(-2)=-2
If your objective is to find the determinant,It is not
necessary to compute the entire matrix cofactors!
you need to determine only the cofactors for the row
or column selected for expansion.
A={5 -4}
{2 -2}
Ac={-2 -2}
{4 5}
|A|=(5)(-2)+(-4)(-2)=-2
PROBABILITY THEORY
A presedential candidate would like to visit seven
cities prior to the next election date.However,
it will be possible for him to visit only three cities.
How many different iteneries can he and his
staff consider?
7!
= ________
(7-3)!
= 7.6.5.4.3.2.1
_______________
4.3.2.1
= 7.6.5
=210
Interesting facts-Divine proportions
The Golden Section is a special ratio that is
approximately equal to 1.618 or (SQR5+1)/2:1.
It has been used in art and architecture for centuries,
and is also found in nature. The Golden Section divides
a line at the point so that the smaller part of the
line to the greater part is the same as the ratio of
the greater part to the whole line.Also known as the
Golden Mean,this proportion is said to be most
pleasing to the eye.The Greeks were intrigued by this
special mathematical relationship.Before that,the
Egyptians certainly had a "sacred ratio";on the Great
Pyramid at Gizeh in Egypt,the ratio of the height of
a face to half of a base is 1.618:1.
Using your brain
Nowdays,with the common use of calculators,some people
feel that the skill in mental arithmetic is no longer
necessary.However,this skill is still important because
it encourages a genuine understanding of numbers.Good
mathematician always look for the simplest methods and
rules and try to reduce complicated calculations down
to a series of basic sums.These methods need practice,
but eventualy become invaluable aids that you use
automatically.
Mathematical genius
Shakantula Devi is an Indian woman of immense
arithmetical ability who demonstrates her amazing
skills in public.It took just a second for her to find
the cube root of 332,812,557 is 693.In 1970 she
calculated the 23rd root of a 201-digit number.The
problem had been devised by Texan students.A computer
took one minute and 13000 instructions to check her
answer and prove her right.
Archimedes
Archimedes(287-212BC) is considered to be the greatest
mathematician of antiquity.He lived in Syracuse(Sicily,
Italy)after studing for a short while in Alexandria,
Egypt.He was highly practical man and many mathematical
inventions are attributed to him,but he placed most
value on the products of his thoughts.Among other
things he developed the laws governing floating bodies,
a method for achieving fantastically large calculations
an accurate estimate of pi and numerous geometrical
formulae.His formula regarding the volume of a sphere
was inscribed on his tombstone after he was killed by
a Roman soldier.
Michael Faraday
Michael Faraday(1791-1867)was an English physicist who
had little formal education.He was a brilliant
experimental scientist and became Professor of Chemistry
at the Royal institution,London.One of Faraday's
dicoveries was the principle of electromagnetic induction
and he made the first electiric motor.Not until the
Scottish mathematician James Maxwell(1831-1879)
formalized these discoveries using mathematical formulae
could physicists build on Faraday's work.
What is zero?
The name zero comes from the Latin zephirum,meaning
empty or blank.The symbol "0" originated in India;
in AD 830 al Khwarizimi explained the system of Indian
numerals including the use of zero,but it was not
translated for use in the West for another 400 years.
Zero continued to puzzle scholars.Was it a number or
a digit?if if it stood for nothing,then surely it did
not need to be included?Leonardo of Pisa,or Fibonacci
(1180-1250),in his book Liber Abaci,provided the answer.
He said that zero can be used as a "place holder" to
seperate columns of figures.It can also represent a
position on a scale.In temperature scales,zero degrees
is a valid reading;it does not mean "no temerature".
Estimating
ALTHOUGH MATHEMATICS is a precise discipline,in pactice
we often need only approximate answers.For example,
companies packaging goods estimate the amount that
consumers are likely to use,and economists estimate
finacial trends.Someone about to cross a road estimates
the speed of oncoming cars so that they know whether
or not to wait before crossing.Often we instinctively
round off a figure to an approximate value.You may say
you are 12 years old rather than 12¼,or that you will
be home in about 20 minutes,not 18 minutes and 42
seconds.Making mistakes with a calculator is easy;to
spot them you should first estimate.
Trial Matric exam(SG)
Develop speed in doing exams
Trial Matric exam(HG)
most important enjoy maths!