Its all about the algebra, inequalities, variation, Graphs and sequences. I hope u find the ppt informarive

1. Algebraic representation and
manipulation

2. Expanding two brackets
(3 + t)(4 – 2t)
To expand or multiply out this expression, multiply every term in the second
bracket by every term in the first bracket.
(3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t)
= 12 – 6t + 4t – 2t2
= 12 – 2t – 2t2
This is a
quadratic
expression.

3. Squaring expressions
Expand and simplify: (2 – 3a)2
(2 – 3a)2
= (2 – 3a)(2 – 3a)
= 2(2 – 3a) – 3a(2 – 3a)
= 4 – 6a – 6a + 9a2
= 4 – 12a + 9a2

4. Transformation of complex formulae
P = 2mx
X = p / 2m
V = u^2 + 2zt
v – 2zt = u^2
root to v – 2zt = u

5. Factorizing
a) 9x2
+ 3x – 18
b) 7p2
+ 21
c) 4w2
+ 2w
a.) 3(3x2
+ x – 6)
b) 7(p2
+ 3)
c) 2w(2w + 1)

6. Evaluation
8^2 – 2^2
= 64 – 4
= 60
8.4^4 – 2.5^4
= 70.56 – 6.25
= 64.21

7. Addition
bd
bcad
d
c
b
a +
=+
xx −
+
2
31
Simplify
)2(
)(3)2(1
xx
xx
−
+−
=
)2(
32
xx
xx
−
+−
=
)2(
22
xx
x
−
+
=

8. Subtraction
bd
bcad
d
c
b
a −
=−
p
t
p 2
2
−Simplify
)2)((
))(()2)(2(
pp
tpp −
=
2
2
4
p
ptp −
=
2
2
)4(
p
tp −
=
p
t
2
)4( −
=

9. Multiplication
bd
ac
d
c
b
a
=×
5
5
3 +
−
×
x
xx
Simplify
)5(3
)5)((
+
−
=
x
xx
)5(3
52
+
−
=
x
xx

10. Division
bc
ad
d
c
b
a
=÷
Simplify
14
4
7
2 yx
÷
y
x
4
14
7
2
×=
)4)(7(
)14)(2(
y
x
=
2
2
y
x
=

11. 7
43
2
=−
xx
Solve
7
)4)(3(
))(3()2)(4(
=
−
⇒
xx
7
12
38
=
−
⇒
xx
7
12
5
=⇒
x
845 =⇒ x
5
84
=⇒ x

12. Algebraic indices

13. Rule 3 : Brackets
(b6
)2 = b6
x b6
= b12 (am
)n
= am x n
Rule 4 : Index of 0
n5
÷ n5
= n5-5
= n0
= 1 a0
= 1

14. Rule 1 : Multiplication
c6
x c4
= c10
am
x an
= am+n
Rule 2 : Division
am
÷ an
= am-n
m5
÷ m7
= m-2

15. Rule 5 : Negative indices
n
n
a
a
1
=−
=−3
a 3
1
a
pq
aa q
p
=
Rule 6 : Fractional indices
p is the power
q is the root

16. Solved examples
a6
x a4
= a10
b5
x b7
= b12
a5
x a3
a4
x a6
= a8
a10
= a-2
c5
x c3
c4
= c8
c4
= c4

17. Simultaneous equations

18. Linear Equations
132 =−x
312 +=x
2
2/4
42
=
=
=
x
x
x

19. Basic steps
Look at equations
Same number of x’s or y’s?
If the sign is different, ADD the
equations otherwise subtract tem
Then have ONE equation
Solve this
Substitute answer to get the other
CHECK by substitution of BOTH
answers

20. Elimination Method
4x - 3y = 14
2x + 3y = 16
We have the same
number of y’s in each
If we ADD the equations,
the y’s disappear
+
6x = 30 Divide both sides by 6
x = 5
A
B
Substitute x = 5 in equation A
20 – 3y = 14
3y = 6
y = 2
Answer
x = 5, y = 2

21. Elimination Method
2x + 3y = 9
2x + y = 7
We have the same
number of x’s in each
If we SUBTRACT the equations,
the x’s disappear
-
2y = 2 Divide both sides by 2
y = 1
A
B
Substitute y = 1 in equation A
2x + 3 = 9
2x = 6
x = 3
Answer
x = 3, y = 1

22. What if NOT same number of x’s or y’s?
5x + 2y = 17
3x + y = 10
-
x = 3
In B
A
B
5 x 3 + 2y = 17
15 + 2y = 17
y = 1
Answer
x = 3, y = 1
If we multiply A by 2 we
get 2y in each
5x + 2y = 17
6x + 2y = 20
B
A

23. …if multiplying 1 equation doesn’t help?
3x + 7y = 26
5x + 2y = 24
-
29y = 58
In B
A
B
5x + 2 x 2 = 24
5x = 20
x = 4
Answer
x = 4, y = 2
Multiply A by 5 & B by 3,
we get 15x in each
15x + 35y = 130
15x + 6y = 72B
A
y = 2
Could multiply A by 2 & B
by 7 to get 14y in each

24. The sum of three consecutive integers is 54. What
are the three integers?
Three consecutive integers can be represented as
n, n + 1, n + 2.
n + (n + 1) + (n + 2) = 54
3n + 3 = 54
3n = 51
n = 17
Simplify left-hand side.
Subtract 3.
Divide by 3.
The three consecutive integers are 17, 18, and 19.
17 + 18 + 19 = 54.
Linear equation

25. The Quadratic
Equation

26. Q
1 x2
+ 4x -2=0
a b c

27. x2
+ 4x -2=0
Start with the quadratic equatioStart with the quadratic equatio
Put in our values for a, b andPut in our values for a, b and c
Tidy upTidy up
You need to do this twice, once
adding and once taking away
You need to do this twice, once
adding and once taking away
and
x = -4.4 (1dp) and x = 0.4 (1dp)x = -4.4 (1dp) and x = 0.4 (1dp)

28. Linear Inequalities
Symbols

29. Solutions continued….
-5 -4 -3 -2 -1 0 1 2 3 4 5
1≤x

30. Linear Programming

31. Sketch a graph of y ≥ 3

32. The shaded area is where y is less than or equal to x + 2

33. Sequences

34. Here is a sequence of numbers.
What number is going
to come next in this
sequence?
0 2 4 6 8 10 12 14 16
Correct it is
the number
18

35. The nth term of an arithmetic sequence is given by:
The last # in the
sequence/or the #
you are looking for
First
term
The position
the term is in
The common
difference
un = a + (n – 1) d

36. Find the 14th
term of the
sequence 4, 7, 10, 13,……
(14 1)+ −4
4 (13)3= +
4 39= +
43=
un = a + (n – 1) d
u14 = 3
The 14th
term in this
sequence is the no. 43

37. a+b+c 4a+2b+c 9a+3b+ca+b+c 4a+2b+c 9a+3b+c
3a + b3a + b
2a2a
5a + b5a + b
Quadratic sequence

38. a+b+c 4a+2b+c 9a+3b+ca+b+c 4a+2b+c 9a+3b+c
3a + b3a + b
2a2a
5a + b5a + b
3 4 11 243 4 11 24
11
66
77 1313
66

39. a+b+ca+b+c
3a + b3a + b
2a2a
33
11
66
6 = 2a6 = 2a
1 = 3a + b1 = 3a + b
3 = a + b + c3 = a + b + c

40. 6 = 2a6 = 2a
1 = 3a + b1 = 3a + b
3 = a + b + c3 = a + b + c
a = 3a = 3
b = -8b = -8
c = 8c = 8

41. Cubic sequences

42. Geometric sequences

43. × 2 × 2 × 2 ×
2
r = 2
1−
= n
n ara
5th term.
a = 1
( ) 1621
15
5 ==
−
a
1, 2, 4, 8, 16 . . .

44. Direct and Inverse Variation

45. Direct Variation
If y varies directly as x, then y = kx.
So if x = 2 and y = 10
Therefore, by substitution
What is the value of k?
10 = k(2).
10 = 2k
5 = k

46. Inverse Varation
Since Direct Variation is Y=kx
(k times x)
then
Inverse Variation is the opposite Y=k/x
(k divided by x)

47. inversely Variation
If y varies inversely as x, then y = k/x.
So if x = 2 and y = 10
Therefore, by substitution
What is the value of k?
10 = k/(2).
10(2) = k
20 = k

48. New Found Knowledge
Invented by =the Greek
mathematician Diophantus
He was also know as the father of
algebra
Year - 1637

49. Questions
1. , what is a, b and c?
- 2x2
+4x -3 =0
2. A table top measures x cm by y cm. Write down an
expression for its area, A.
3.
4. Evakutate - 7^2 + 12^2
●Solve 2x+3>x+5