SOLUTION OF QUADRATIC EQUATIONS BY FACTORISATION
SOLVE THE FOLLOWING EQUATION:
1.1 x�-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
x�-x=6
x�-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 x�+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)�+(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
______ + ______ = ______ - ______
x�-4 x�+x-2 2-x 4-x�
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)
x�+2x-3+x-2=-x�-x+2+7x-7
2x�-3x=0
x(2x-3)=0
x=0 or x= 3
___
2
2.2 x-3 5 4
______ - _______ = _______
x�+3x+2 x�-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)
x�-5x+6-5x-5=-4x�+16
5x�-10x-15=0
x�-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-x�
2.9 2 2 1 3
_______ = ______ + _________ + ________
x�+3x+2 1-x� x�+3x+2 x�+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: x�-2x=18- 45
_____
x�-2x
Let x�-2x=K
Then: K=18K-45
K�=18K-45
(K-15)(K-3)=0
K=15 or K=3
x�-2x=15 or x�-2x=3
x�-2x-15 or x�-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
x� x
3.3 6y-2+y-1-2=0
3.4 (x�-3x)�-20=8(x�-3x)
3.5 (x�+3x)�-2(x�+3x)-8=0
3.6 y�-y-3= 9
__________
y�-y-3
3.7 (x�-2x)�-2(x�-2x)-3=0
3.8 (x�-5x)�=36
3.9 (2x�+5x)�-10x�-25x-14=0
3.10 x�-5x+2- 4
________ =0
x�-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=x�
x�-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(x�+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)=x�
then f'(x)=2.1x2-1
f'=symbol for the first differential
[(exponent)(coeficient)
x(variable power-1)]
2. If y=3x�=2x-2
y'=(2)(3)x2-1
y'=6x-4x-3
3. If f(x)=16x�-32x�
Dx=
(2)(16)x2-1-(3)(32)x3-1
Dx=32x-96x�
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)=x3-4x�
To find gradient formula f'(x)=3x�-8x
when x=2 f'(x)=3(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=2-4x�+x3at the point where x=1
To find gradient formula y'=-8x+3x�
Gradient when x=1: y'=-8(1)+3(1)�
=-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=x3-4x�+2
y=(1)3-4(1)�+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)
=(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
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