# Chapter 04: Algebraic Formula & Application

## Keep in Mind!

• Read The Text Book Thoroughly
• Think Why Formulae Are Such
• DO NOT MEMORIZE Formulae

## Formulae

### MEMORIZE FEEL

#### Can you translate them into English?

• $$(a+b)^2 = a^2+2ab+b^2$$
• $$(a-b)^2 = a^2-2ab+b^2$$
• $$a^2-b^2=(a+b)(a-b)$$

#### Corollary

• $$a^2+b^2= ?$$
• $$a^2+b^2= ?$$ (in another form)
• $$(a+b)^2 = ?$$ (in terms of) $$(a-b)^2$$
• $$(a-b)^2 = ?$$ (in terms of) $$(a+b)^2$$
• $$(a+b)^2+(a-b)^2 = ?$$
• $$(a+b)^2+(a-b)^2 = ?$$
• $$ab=?$$

## Relationship between (a+b)2 and (a-b)2

$\begin{eqnarray} (a+b)^2 &=& a^2+2ab+b^2 \nonumber \\ &=& a^2-2ab+b^2 + 2ab + 2ab \nonumber \\ &=& (a-b)^2 +4ab \nonumber \\ \end{eqnarray}$

• Find (a-b)2 similarly

• (2x+y)
• (5m-3p)
• ax+b+2

## 2(a)

Simplify

$$(x+y)^2+2(x+y)(x-y)+(x-y)^2$$

## 2(d)

Simplify

$$(8x+y)^2 - (16x+2y)(5x+y) + (5x+y)^2$$

• $$(8x+y)^2 - 2(8x+y)(5x+y) + (5x+y)^2$$

## 3(a)

Find product using formula

(x+7)(x-7)

## 3(b)

Find product using formula

(5x+13)(5x-13)

## 3(i)

(ax-by+cz)(ax+by-cz)

• {ax-(by-cz)}{ax+(by-cz)}
• $$(ax)^2-(by-cz)^2=a^2x^2-b^2y^2+2bycz-c^2z^2$$

## Application of $$x^2+x(a+b)+ab$$

$\begin{eqnarray} Expression (x+3)(x+4) \nonumber \\ &=& \\ \end{eqnarray}$

## 3(l)

Formula: (x+a)(x+b) = x^2+x(a+b)+ab

## Find Value (5)

If $$x-\frac 1 x=3$$, $$x^2+\frac 1 {x^2}=?$$

Solution 01
Alternative

## Find Value (6)

$$a+\frac 1 a =4, a^4+\frac 1 {a^4}=?$$

## Find Value (7)

If m = 6 and n = 7

$$16(m^2+n^2)^2+56(m^2+n^2)(3m^2-2n^2)+49(3m^2-2n^2)^2$$

## Find Value (8)

$$a-\frac 1 a=m$$

Show $$a^4+\frac 1 {a^4}=m^4+4m^2+2$$

## Express as difference of squares (13a)

1. (5 p - 3q)( p + 7q)
2. (5x - 13)(5x + 13)
Solution

Solution (ii)

• $$(5x)^2-(13)^2$$

## Find Value (9)

$$x-\frac 1 x=4$$; Prove $$x^2+(\frac 1 x)^2=18$$

$\begin{eqnarray} x-\frac 1 x=4 \nonumber \\ &\Rightarrow& x^2-2 + \frac 1 {x^2} = 16 \nonumber \\ &\Rightarrow& x^2+ \frac 1 {x^2} =16+2=18 \nonumber \\ \end{eqnarray}$

# Chapter 4.2

## Formulae

1. $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$
2. $$(a-b)^3=a^3-3a^2b+3ab^2-b^3$$
3. $$a^3+b^3=?$$ (Find from above)
4. $$a^3-b^3=?$$ (Find from above)

• $$a^3+b^3=(a+b)^3-3ab(a+b)$$ (Rearrange)
• $$a^3-b^3=(a-b)^3+3ab(a-b)$$ (Rearrange)
• $$a^3+b^3=(a+b)(a^2-ab+b^2)$$
• $$a^3-b^3=(a-b)(a^2+ab+b^2)$$

## Find Cubes

1. 2a+5b
2. $$a^2b^2-c^2d^2$$
3. $$(2m+3n-5p)^2$$
• 1.$$8a^3+6a^2b+150ab^2+125b^3$$

## Simplify

1. $$(3x+y)^3+3(3x+y)^2(3x-y)+3(3x+y)(3x-y)^2+(3x-y)^3$$
2. $$(6m+2)^3-3(6m+2)^2(6m-4)+3(6m+2)(6m-4)^2+(6m-4)^3$$
1. $$216x^3$$
1. 216

## Find Value(s)

1. If a+b=8 and ab=15, $$a^3+b^3=?$$
2. 2x+3y=13 and xy=6, $$8x^3+27y^3$$
3. $$x-2y=3, x^3-8y^3-18xy=?$$
4. $$4x-3=5, 64x^3-27-180x=?$$
5. $$x=5, x^3-12x^2+48x-64=?$$
6. $$x+\frac 1 x = 4, x^3+\frac 1 {x^3}=?$$
7. $$a^2-1-5a=0, a^3-\frac 1 {a^3}=?$$
• $$152$$
• $$(2x+3y)^2-3 \cdot 2x \cdot 3y \cdot (2x+3y) = 13^2-3 \cdot 6 \cdot 6 \cdot 13 = 793$$
• $$(x-2y)^3+3 \cdot x \cdot 2y (x-2y) - 18xy = 3^3+18xy-18xy=27$$
• $$125$$
• $$x^3-3x^2 \cdot 4 + 3x\cdot 4^2-4^3=(5-4)^3=1$$
• $$4^3-3 \times 4 = 64-12 = 52$$
• $$a-\frac 1 a = 5, \therefore Answer=5^3+3 \times 5=140$$

## Find Products Using Formula

• $$a^3+b^3=(a+b)(a^2-ab+b^2)$$
• $$a^3-b^3=(a-b)(a^2+ab+b^2)$$

Find

1. $$(a^2+b^2)(a^4-a^2b^2+b^4)$$
2. $$(ax-by)(a^2x^2+abxy+b^2y^2)$$
3. $$(x^2+a)(x^4-ax^2+a^2)$$
4. $$(x+a)(x^2-ax+a^2)(x-a)(x^2+ax+a^2)$$
5. $$27x^4-8xy^3$$
6. $$27a^3-8$$

• $$a^6+b^6$$
• $$a^3x^3-b^3y^3$$
• $$x^6+a^3$$
• $$x^6-a^6$$
• $$x(27x^3-8y^3)=\cdots=x(3x-2y)(9x^2+6xy+4y^2)$$
• $$(3a)^3-2^3=\cdots$$

## Find Values

1. a+b=3, ab =2; $$a^3+b^3=?$$
2. x+y=4, $$x^3+y^3+12xy=?$$
3. $$a+\frac 1 a = 7, a^3+\frac 1 {a^3}=?$$
4. m = 2; $$27m^3+54m^2+36m+8=?$$
5. a = -3, b = 2; Find $$8a^3+36a^2b+54ab^2+27b^3$$
6. $$a^2+b^2=c^2; a^6+b^6+3a^2b^2c^2=?$$
7. $$a^2-a-6=0, a^3+\frac 1 {a^3}=?$$
• $$3^3 - 3 \times 2 \times 3 = 9$$
• $$7^3-3 \times 7$$ = 343-21=322
• $$(x+y)^3-3xy(x+y)+12xy=4^3=64$$
• $$(3m)^3+3\cdot (3m)^2 \cdot 2 + \cdots = (3m+2)^3=8^3=512$$
• $$=\cdots = (2a+3b)^3=(-6+6)^3=0$$
• $$c^6$$
• $$a^2-a-6=0 \Rightarrow a^2-3a+2a-6=0 \Rightarrow a=3,-2$$ $$\Rightarrow a+\frac 1 a = 3 + \frac 1 3 = \frac {10} 3$$ or $$\cdots$$

# Chapter 4.3

## Resolve into Factors

1. $$x^2-15x+54$$
2. $$x^2-23x+132$$
3. $$2x^2+9x+10$$
4. $$8x^2+42x+27$$
5. $$16y^2-a^2-6a-9$$
6. $$(x-y)^3+z^3$$
7. $$(x^2-x)^2+3(x^2-x)-40$$
8. $$x^2+(3a+4b)x+(2a^2+5ab+3b^2)$$
9. $$x^2-x-(a+1)(a+2)$$
10. $$ax^2+(a^2+1)x+a$$
11. $$a^3-3a^2b+3ab^2-2b^3$$
1. (x-6)(x-9)
1. (x-12)(x-11)
1. (x+2)(2x+5)
1. $$(36,6) \Rightarrow (4x+3)(2x+9)$$
1. $$(4y)^2-(a+3)^2=()()$$
1. Use $$(a+b)(a^2-ab+b^2)$$
1. Assume $$(x^2-x)^2=a$$ or continue directly.
1. $$x^2+(3a+4b)x+(2a+3b)(a+b)$$
1. $$x^2-(a+2)x+(a+1)x-(a+1)(a+2)$$ or let a+1=p
1. $$ax^2+a^2x+x+a \rightarrow ax(x+a)+1(x+a) \rightarrow ()()$$
1. $$a^3-3a^2b+3ab^2-b^3-b^3 \rightarrow (a-b)^3-b^3 \rightarrow \cdots (a-2b)(a^2-ab+b^2)$$

# Chapter 4.3: LCM & HCF

## What LCM & HCF they Mean

LCM = Lowest Common Multiple

Numbers: 12, 18

Multiples of 12: 12, 24, 36, 48, 60

Multiples of 18: 18, 36, 54, 72, 90

Lowest Common Multiple, LCM = 36

12 = $$2 \times 2 \times 3$$

18 = $$2 \times 3 \times 3$$

LCM = $$2 \times 2 \times 3 \times 3 = 36$$

HCF

## Find LCM & HCF

1. 9abc, 12abc, 15abc
2. $$20x^3y^2a^3b^4, 15x^4y^3a^4b^3, 35x^2y^4a^3b^2$$
3. $$x^2-3x, x^2-9, x^2-4x+3$$
4. Can HCF be greater than LCM?
5. $$a^2b(a^3-b^3), a^2b^2(a^4+a^2b^2+b^4), a^3b^2+a^2b^3+ab^4$$
6. $$a^3-1, a^3+1, a^4+a^2+1$$
1. 180abc ,3abc
1. $$420a^4b^4x^4y^4, 5a^2b^2x^2y^2$$
1. x(x-3), (x+3)(x-3), (x-1)(x-3)
1. Clue: can factor of a number be greater than itself?
Can a multiple be smaller?
1. $$Factors: a^2(a-b)(a^2+ab+b^2),$$
$$a^2b^2(a^2+ab+b^2)(a^2-ab+b^2), ab^2(a^2+ab+b^2)$$
• 6 Factors: $$(a-1)(a^2+a+1), (a+1)(a^2-a+1)$$
$$(a^2+a+1)(a^2-a+1)$$

## Can HCF be > LCM

Consider 0, 9

Factors

$$0 \rightarrow 0, 1, 2, 3, \cdots$$ since $$0 \times 1 = 0, 0 \times 2 =0$$

$$\therefore$$ all numbers are factors of 0.

$$9 \rightarrow 1, 3, 9$$

Multiples

We get multiples by multiplying the numbers by $$1, 2, 3, \cdots$$

$$0 \rightarrow 0, 0, 0 , \cdots$$

$$\therefore$$ only 0 is the mutliple of 0.

$$0=9 \rightarrow 9, 18, 27 , \cdots, 0$$ (0 is a multiple of any number)

HCF = 9

LCM = 0

# Ch 5.1: Algebraic Fractions

## Express in Lowest Form

1. $$\frac{4x^2y^3z^5}{9x^5y^2z^3}$$
2. $$\frac{x^3y+xy^3}{x^2y^3+x^3y^2}$$
3. $$\frac{x^2-6x+5}{x^-25}$$
1. $$\frac{4yz^2}{9x^3}$$
1. $$\frac{x^2+y^2}{xy(x+y)}$$
1. $$\frac{x-1}{x+5}$$

## Express in Common Denominator

1. $$\frac{x-y}{xy}, \frac{y-z}{yz}, \frac{z-x}{zx}$$
2. $${a-b} \over {a+2b}$$, $${2a+b} \over {a^2-4b^2}$$
1. $$\frac{xz-yz}{xyz}, \frac{xy-xz}{xyz}, \frac{yz-xy}{xyz}$$
1. $${(a-b)(a-2b)} \over {a^2-4b^2}$$, $${2a+b} \over a^2-4b^2$$

Divide each denominator by LCM and multiply the quotient with each term.

1. $$\frac 2 {x^2y-xy^2}, \frac 3 {xy(x^2-y^2)}, \frac 1 {x^2-y^2}$$
2. $$\frac 1 {x^2-1}+\frac 1 {x^4-1}+ \frac 1 {x^8-1}$$
1. $$\frac{2x+xy+2y+3}{xy(x-y)(x+y)}$$
1. $$\frac{x^6+2x^4+x^2+6}{x^8-1}$$

## Subtract

1. $$\frac x {(x-y)^2}-\frac{x+y}{x^2-y^2}$$
2. $$\frac 1 {x-y} - \frac{x^2-xy+y^2}{x^3+y^3}$$
1. $$\frac y {(x-y)^2}$$
• $$\frac{2y}{x^2-y^2}$$

## Simplify

1. $$\frac 1 {1-a+a^2}-\frac 1 {1+a+a^2}-\frac 1 {1+a^2+a^4}$$
2. $$\frac 1 {x-y} - \frac 2 {2x+y} + \frac 1 {x+y} - \frac 2 {2x-y}$$
1. 0
1. $${6xy^2} \over {(x^2-y^2)(4x^2-y2)}$$

## Arranging Soldier Problem

Arrange 10 soldiers in 5 rows so that each row contains 4 soldiers.

Solution (Click to see)

# Chapter 5.2

## Multiply

1. $$\frac{x-1}{x+1}, \frac{(x-1)^2}{x^2+x}, \frac{x^2}{x^2-4x+5}$$
2. $$\frac{1-b^2}{1+x}, \frac{1-x^2}{b+b^2}, 1+\frac{1-x}{x}$$
1. $$\frac{x(x-1)^3}{(x+1)^2(x^2-4x+5)}$$
1. $$\frac{(1-b)(1-x)}{bx}$$

## Divide

1. $$\frac{x^2-7x+12}{x^2-4}, \frac{x^2-16}{x^2-3x+2}$$
1. $$\frac{(x-3)(x-1)}{(x+2)(x+4)}$$

## Simplify

1. $$(\frac a b + \frac b a + 1) \div (\frac{a^2}{b^2}+\frac a b + 1)$$
2. $$(\frac 1 x + \frac 1 y) \times (\frac 1 x - \frac 1 y)$$
1. $$\frac b a$$
1. $$\frac{y^2-x^2}{x^2y^2}$$

# Chapter 6: Equation

## Creative Question -01

Observe the pair of equations below:

• $$x + ay = b ... (i)$$
• $$ax-by=c... (ii)$$
1. Of which equations is the solution $$(b,0)$$?
2. If $$a=1, b=2, c = 3$$, solve the pair of equations.
3. Solve the given pair of equations by using eliminations method.

## Creative -02

5 years ago the ratio of ages of father and son was 7:1 and after 10 years, the ratio would be 5:2

1. Form two equations with two variables.
2. Find the present ages of father and son.
3. Form an equation with one variable and calculate the present age of father and son.

## Creative -03

There are two numbers. The sum of thrice of the first number and the second number is 17 and sum of the first and thrice of the second number is 19.

1. Form two equations.
2. Solve by substitution method.
3. Solve by means of graph (without using (2)).
• $$3x+y=17$$
• $$x+3y=19$$
• $$3x+9y=57$$
• 8y = 40
• y = 10

## Number-digit

If 7 is added to the sum of digits of a two-digit number, the sum is thrice the digit in tens place. But if 18 is subtracted from the number, the digits switch places. Determine the number.

• x = ones place digit, y = tens place digit
• Number: $$x + 10y \space [\because 21 = 1 + 2\times 10$$]
• $$x+y+7=3y \rightarrow x-2y=-7$$
• $$x+10y-18=y+10x \Rightarrow -9x +10y = 18$$
• $$y-x = 2$$
• (3,5)

# Statistics

## What is Statistics

Statistics has three meanings:

1. Data (table or a series of values)
Light speed experiment
Expt Run Speed
001 1 1 850
002 1 2 740
003 1 3 900
004 1 4 1070
005 1 5 930
006 1 6 850
1. Plural of statistic (formula)
2. Method of analyzing and predicting data

## Data

Data: Information expressed in numbers (usually) (NOT a GOOD Definition)

• The quantities, characters, or symbols on which operations can be performed by a computer
• Factual information used as a basis for reasoning, discussion, or calculation (~ Merriam-Webster)
• Example: 10, 23, 6, 16, 19, 25, 14, 12, 22, 28, 4, 21, 2, 7, 17

## Types of Data

1. Primary Data: Collected directly
2. Secondary Data: Fetched from someone else
Examples

## Unorganized and Organized Data

40, 39, 31, 38, 40, 40, 34, 39, 31, 38, 37, 30, 31, 37, 35, 37, 36, 35, 39, 39

x Freq
30 1
31 3
34 1
35 2
36 1
37 3
38 2
39 4
40 3

## Frequency Distribution

X = 11, 15, 16, 18, 20, 22, 25

Class Tally Frequency
11-15 || 2
16-20 ||| 3
21-25 || 2

## Construction

Range = (Highest value - Lowest value) + 1

X = 11, 15, 16, 18, 20, 22, 25

Range of X = ?

Class Interval

$$(11-15) \rightarrow (15-11+1)= 5$$, not 4

• $$(20-24) \rightarrow ?$$
• Number of class = $$\frac{Range}{Interval}$$
• Interval = ?

## Example of Frequency Distribution

X = 32, 20, 34, 17, 15, 40, 5, 18, 44, 28, 49, 27, 8, 29, 45, 39, 3, 35, 46, 37, 50, 36, 2, 4, 7, 24, 42, 31, 19, 14

Distribution

Range = ?

Let, class interval = 5

Number of class = ?

• Now, construct

## Interpretation

Class Frequency
11-15 2
16-20 5
21-25 9
26-30 10
31-35 3
• What have you known from this frequency distribution?
• What is the benefit of organizing?

## Histogram

• Make sure class intervals are continuous

Continuous or exclusive: (10-15); (15-20); (20-25) Discontinuous/Inclusive: (10-14); (15-19)

• If discontinuous $$\rightarrow$$ convert
• Add 0.5 to upper limit and subtract 0.5 from lower limit
$$\downarrow$$

## Histogram Example

Interval Frequency
20-30 5
30-40 12
40-50 30
50-60 40
60-70 20
70-80 13
80-90 3
90-100 2

Write its interpretation in 3-5 sentences.

## Make a Histogram

Class Interval Continuous CI Frequency
11-20 10
21-30 20
31-40 35
41-50 20
51-60 15
61-70 10
71-80 8
81-90 5
91-100 3

Histogram