leaving cert algebra
DESCRIPTION
LEAVING CERT ALGEBRA. SUMMARY OF THE SECTIONS IN L.C. ALGEBRA. 1. SIMPLIFY. Squaring Rule. Division in Algebra. Surds. Common (grouping). Quadratic. 2. FACTORS. Difference of two squares. Sum and difference of two cubes. Linear. Quadratic. Cubic. 3. FUNCTIONS AND EQUATIONS. - PowerPoint PPT PresentationTRANSCRIPT
LEAVING CERT ALGEBRALEAVING CERT ALGEBRA
1. SIMPLIFY
Simplifying Expressions
Substitution
Squaring Rule
Division in Algebra
SUMMARY OF THE SECTIONS IN L.C. ALGEBRA
Surds
2. FACTORS
Common (grouping)
Quadratic
Difference of two squares
Sum and difference of two cubes
3. FUNCTIONS AND EQUATIONS
Linear
Quadratic
Cubic
Simultaneous
2 unknowns
Non-linear
Express in terms of
5. INEQUALITIES
Linear
Quadratic from Graphs
Single
Double
6. INDICES
SECTION 1 SIMPLIFYING
2Example 1. Simplify ( 3)( 2 )x x x
3 2 22 3 6x x x x
2 2( 2 ) 3( 2 )x x x x x
2( 3)( 2 )x x x
3 25 6x x x
Simplifying Expressions
2Example. 2 Simplify (3 -1)(1- 2 ) -5( - 1) x x x x
2 23 - 6 -1 2 -5 5 -5x x x x x 2-11 10 - 6x x
2(3 -1)(1- 2 ) --5( 1)x x x x
x 3 4Example 3. Simplify (Express as a single fraction)
2 5
x
3 4
2 5
x x
When dealing with fractions always get a common denominator.
5 3 2( 4)
(2)(5)
(x ) x
5 15 2 8
10
x x
3 23
10
x
2
3 5Example 4. Simplify (Express as a single fraction)
2 2x x
2
3 5
2 2x x
When dealing with fractions always get a common denominator.
2
3 5
2 2x x
2
2
3 2 5( 2)
( - 2)( 2)
(x ) x
x x
2
2
3 6 5 10
( - 2)( 2)
x x
x x
2
2
3 5 16 ( - 2)( 2)
x x
x x
Substitution2 2 -3 1
Example 1. Find the value of when 3 4 2
x xx
1 Substitute in for every .
2( ) 2 2( ) 3
3 4
x
12
12
122 1 3
3 4
2.5 2
3 4
2.5 2
3 4
0.833 0.5
1.333
Calculator
1 1Example 2. Find the value of when and 2
1 2
a ba b
a b
1
Substitute for every and 2 for every 2
( ) ( ) 1
( ) ( ) 1
a b
1
2 21
2 2
12
12
2 1
2 1
12
12
1 2
3
12
12
3
1 7
2 2
1 2
2 7
1
7
Calculator
Squaring2Example 1. Simplify ( )x y
2( )x y
( )( )x y x y
2x xy yx2y
2 22x xy y
x ( x + y ) + y ( x + y )
Simplifying SurdsWhen simplifying surds we use the following :
2224248 Example:
.
baab 1.
b
a
b
a2.
5
3
25
3
25
3Example:
bbb 3. 555 Example:
Only like surds can be added or subtracted.4. 3232 Example:
3 7 7 2 7 Example:
5. Multiplying surds 2 3 5 2 10 6 Example:
Example: (3 2)(3 2) 9 3 2 3 2 2 2
= 9 2
= 7
6. Irrational Denominator Example: 2
Simplify 3
Irrational Denominator
2 3 2 3
33 3 Rational Denominator
Example: 3
Simplify 1 3
Irrational Denominator
3 1 3
1 3 1 3
3 3 3
1 3
3 3 3
2
Rational Denominator
2Example 1. Simplify
1x
2 1
1 1
x
x x
2 1
1
x
x
1Example 2. Simplify
1
x
x
1 1
1 1
x x
x x
1
1
x x x x
x
SECTION 2 FACTORSType 1 Common Factor (Grouping)
2Example 1. Factorise 3 3x x xy y
2 3 3 Group like termsx x xy y 2 ( 3 ) ( 3 )x x xy y
( 3) ( 3)x x y x
( 3)( )x x y
Type 2 Quadratic Factors
2Example 1. Factorise 3 2x x
Method 1 Brackets Method 2 Big X Method 3 Guide Number
2 3 2x x
( )( )x x2 1
( 2)( 1)x x
2 3 2x x x
x
2
1
( 2)( 1)x x
21 3 2x x
Guide Number ( )( )1 2 = 2Factors for Guide Number
2 1 21 3 2x x
21 2 1 2x x x 2(1 2 ) (1 2)x x x ( 2) 1( 2)x x x ( 2)( 1)x x
2Example 2. Factorise 12 7 10x x
Method 1 Brackets
212 7 10x x
( )( )4x 3x5 2
(4 5)(3 2)x x
4x
3x
5
2
Guide Number ( )( ) =12 10 120
Factors for Guide Number
120 1, 60 2, 40 3
30 4, 24 5, 2
15 8
0 6
, 12 10.
2(12 15 ) (8 10)x x x 3 (4 5) 2(4 5)x x x
( 2)( 1)x x
Method 2 Big X
212 7 10x x
(4 5)(3 2)x x
Method 3 Guide Number
2 712 10x x
212 15 8 10x x x
212 7 10x x
Type 3 Difference of Two Squares2 2 = ( )( )x y x y x y
2Example 1. Factorise 16 25x
216 25x 2 2(4 ) (5)x
(4 5)(4 5)x x
2Example 2. Factorise 1 16a
21 16a2 2(1) (4 )a
(1 4 )(1 4 )a a
Quadratic EquationsQuadratic equations have two solutions (roots).
2Example 1. Solve for if 3 2 0x x x Method 1 Using Factors
2 3 2 0x x ( 2)( 1) 0x x 2 0 or 1 0x x
2 or 1x x
Method 2 Using Quadratic Formula2 3 2 0x x
2 4
2
b b acx
a
1a 3b 2c 2( ) ( ) 4( )( )
2( )x
13 23
1
3 9 8 3 1
2 2x
3 1 3 1 or
2 2x x
1 or 2x x
2Example 2. Solve for if 4 0x x x
Method 1 Using Factors2 4 0x x ( 4) 0x x
0 or 4 0x x
0 or 4x x
Method 2 Using Quadratic Formula2 4 0x x
2 4
2
b b acx
a
1a 4b 0c 2( ) ( ) 4( )( )
2( )x
14 04
1
4 16 0 4 4
2 2x
4 4 4 4 or
2 2x x
4 or 0x x
2Example 2. Solve for if 4 0x x
Method 1 Using Factors2 4 0x
( 2( 2) 0x x 2 0 or 2 0x x
2 or 2x x
Method 3 Using Quadratic Formula2 4 0x
2 4
2
b b acx
a
1a 0b 4c 2( ) ( ) 4( )( )
2( )x
10 40
1
0 0 16 0 4
2 2x
0 4 0 4 or
2 2x x
2 or 2x x
Method 2 Using 2 4 0x 2 4x
4x 2x
2 or 2x x
Example 1. Solve 2 5 3
3 - 2 7
x y
x y
72y- 3x
35y 2x
5310y-15x
610y4x
1941
x
41 x19
195-
y
Method 1
3 52 3 5
2
3 2 7
yx y x
y
Method 2
Simultaneous Equations
Type 1 Two Linear.
3 3 5 4 149 15 4 14 19 5
5
19
53 5
19
2
41
19
y yy y
y
y
x
x
3 5
2
y
2 2Example 1 Solve 2 - 14
1
x y
x y
Type 2 One Linear, One Non Linear.
2 22 - 14
1
x y
x y
1x y 2 22( ) - 14y 1y
2 22( +1+2 ) 14y y y
2 22 +2 + 4 14y y y 2 + 4 12 0y y
( 6)( 2) 0y y
6 0 or 2 0y y
6 or 2y y 5x 3x
This rearranging is often called “changing the subject of the formula” or “express in terms of ”.
Express in terms of.
Example 1 Make the subject of the formula 3 y ax y z
3 ax y z
3 ax ax y z ax
3 y z ax
3
3 3 3
y z ax
3 3
z axy
2Example 2 If 2 express in terms of , and . v u as a s u v
2 2v u as
22 2( ) 2v u as
2 2 2v u as
2 2 2 2 2v u u u as 2 2 2v u as
2 2 2
2 2 2
v u as
s s s
2 2
2 2
v ua
s s
SECTION 5 INEQUALITIES
-1x
-33x
sign. a also is symbol inequality The
signs
33
693
963
Change
x
x
x
Single.
Example 1 Show on a numberline the solution set of 3 6 9, .x x R
1 0- 2 - 1- 3- 4 2 3 4 5
Double.
Example 1 Show on a numberline the solution set of 5 3 1 7, .x x R
3 1 1 7 1x 3 1 7x
3 6x
2x
bits. twointo upSplit
5 5 3 1 5x 0 3 6x
6 3x 2 x
2x
5 3 1x
1 0- 2 - 1- 3- 4 2 3 4 5
5 3 1 7x
5 3 1 7x
5 3 1 7x
’04, LCO, Paper 1
3( -8 -5)
3(-13)
Value is -39
3(2(-4) – 5)
2. (a) Find the value of 3(2p – q) when p = -4 and q = 52. (a) Find the value of 3(2p – q) when p = -4 and q = 52. (a) Find the value of 3(2p – q) when p = -4 and q = 5
Method: Get a common denominator
1
1
x
x
( ) 1 1
(1)
x x
x
(1) xcommon denominator =
1x
x
Method: Use previous answer and cancel
2 1
1
xx
x x
2 1
1
x x
x x
2
Method: Use previous answer and solve
2 3x
2 3 x
5 x
Method: Isolate x
ax b c
Step 1: Take b from both sides
ax c b
c bx
a
Step 2: Divide both sides by a
Method: Solve the inequality and then select all appropriate integers for the set
3 2 4x
3 4 2x 3 6x
2x
............, 3, 2, 1,0,1,2
1 2 3 4 5 6 70-1-2-3-4-5 8-6-7-8
Remember the set of
integers Z contains all
positive and negative whole
numbers and zero.
A =
1 35
2
x
1 3 5 2x 1 3 10x
3 10 1x 3 9x
3x
3x
B 2, 1,0,1,2,3,4,............
1 2 3 4 5 6 70-1-2-3-4-5 8-6-7-8
Multiply both sides by 2
Take 1 from both sides
Divide both sides by 3
Multiply both sides by -1
Remember this will change the direction of the inequality
List the solution set
Or show the solution set on the number line
............, 3, 2, 1,0,1,2 2, 1,0,1,2,3,............
A B= 2, 1,0,1,2
A B=
1 2 3 4 5 6 70-1-2-3-4-5 8-6-7-8
6 12x 5 5x
6 12 5 5x x
6 5 12 5x x
7x
3(2 4)x 5( 1)x
2006 Paper 1: Question 2
Solve simultaneously between Equation 1 and Equation 2 to find the values of a and b
2 15 Equation 1 a b
12 Equation 2 a b
3 27a
( 1592 ) b
18 15b
15 18b
3b
9a
2006 Paper 1: Question 3
2
ab c 3a 2
3b 1c
23( )( )3 (
2
1)
2 1
2
1
2
2 2
2 10
20
x y
x y
10 2x y
2 2( 20 2 )1 0y y 2 2100 4 4 200 yy y
2 80 05 40y y 2 08 16y y
4) 0( 4)(y y
4 0 or 4 0y y
One solution 4y
Therefore line is a tangent.
10 2 )4(x
2x
3 2Simplify by multiplying both sides by ( 2) ( 2) ( 2)
2
xx x x x
x
2 2 3 2x x x 2 4 3 0x x
2 4
2
b b acx
a
1a 4b 3c
24 4( ) ( ) 4( )( )
2( )
1 3
1x
4 16 12
2x
4 28
2
4 2 7
2
x
x
2 7x