Algebraic Operations

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Algebraic Algebraic OperationsOperations

Removing Brackets of the type 2(x-1)

Pair of brackets (x+3)(x+4)

Removing Brackets of the type y(3+a)Removing Brackets of the type -2(x+1)

Equations and brackets 3(x-4) = 2(x-1)

More equations (x+2)2 =x2 +64Apr 22, 2023Apr 22, 2023 Created by Mr. Created by Mr.

[email protected]@mathsrevision.com

Squaring Brackets (x+4)2

Apr 22, 2023Apr 22, 2023

Starter QuestionsStarter Questionsww

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(a) (-3a) x 5a = -15a (a) (-6x) x (-7y) =-42xy

Q2. Are the 2 answers the same ?(a) (-3h)2 = (b) -(3d)2 =

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Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Q3. Spilt £64 pounds into the ratio 7:9

Q4. Explain why 0.0675 is equal to 6.75x10-2

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1. To show how to multiply out (remove) a single bracket with a positive number outside the bracket.

1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket and a positive bracket and a positive number outside the number outside the bracket.bracket.

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2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket.bracket.


Removing aRemoving aSingle BracketSingle Bracket

22 Apr 202322 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.www.

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Simplifying Algebraic Expressions

Reminder !We can only add and subtract “ like terms “

x x x x 4x9 7p p 2p

10 6 1w 3 6a b a b

2 2 2x x x

2 7a b9 6w

23x

22 Apr 202322 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.www.

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Simplifying Algebraic Expressions

Reminder !Multiplying terms

7 a 7a5w 5w

2 4m m b b 2b

28m( NOT 2b )

( NOT 8m )

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3(b + 5) =3b + 15Example 1

4(w - 2) =4w - 8Example 2

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2(y - 1) =2y - 2Example 3

7(w - 6) =7w- 42Example 4



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8(x + 3) =8x+ 24Example 5

4(3 -2m) = 12 - 8mExample 6



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Example 7 : Write down an expressionfor the amount indicate belowin two difference ways.

3(x + 3) =3x + 9



x+3x+3

x+3

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Example 8 : Write down an expressionfor the amount indicate belowin two difference ways.

5(y - 2) = 5y - 10



y-2

y-2 y-2

y-2 y-2

22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Now try Ex 2.1

Ch3 MIA (page 43)

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Apr 22, 2023Apr 22, 2023


w.m

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(a) -3y x 5y =-30y (b) -6q x (-4q) = 24q2

Q2. Is the following true

-2a( b – a) = -2ab +4a

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Q3. Write down the two numbers that multiplyto give 8 and subtract to give 2.

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1. To show how to multiply out (remove) a single bracket with a letter outside the bracket.

1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket with a letter bracket with a letter outside.outside.

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2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket with a letter bracket with a letter outside the bracket.outside the bracket.



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a(b + 5) =ab + 5aExample 1

p(w - 2) =pw - 2pExample 2



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y(y - 1) = y2 - yExample 3

7w(w - 3) =7w2 - 21wExample 4



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a(a2 + 3b) =a3 + 3abExample 5

y2(3y -2c) =3y3 - 2cy2

Example 6




Now try Ex 2.2

Ch3 MIA (page 44)

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Apr 22, 2023Apr 22, 2023


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(a) 3y(x - y) = (b) 6q2 (2 - 4q) =

Q2. Explain your working to show that if we split 24 into the ratio 1:5 the answer is 4:20

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Q3. Writing out in full 5.2 x 10-3 to get 0.052Is the correct?

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1. To show how to multiply out (remove) a single bracket with a negative outside the bracket.

1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket with a negative bracket with a negative outside.outside.

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2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket with a negative bracket with a negative outside the bracket.outside the bracket.



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-(x + y) = -x - yExample 1

-5(a - 7) =-5a + 35Example 2



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7 + 3(4 - y) =7 + 12Example 3

9 - 3(8 - y) =9 - 24Example 4



- 3y= 19 - 3y

Tidy Up

+ 3y

Tidy Up

= -15 + 3y

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A = 10(x – 2)

Example 9



Find my Area (x - 2)

10Find my Areax

12

A = 12xA = L x B A = L x B

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Example 9 : Find area of the orange border



(x - 2)10x

12Area =Big Rec – Small Rec

= 12x – 10(x - 2)= 12x – 10x + 20= 2x + 20


Now try Ex 3.1

Ch3 MIA (page 44)

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Apr 22, 2023Apr 22, 2023


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given the first has area 7w - 14

Q2. Split 48 into the ratio 1:3

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Q3. In standard form 18294000 is 1.8x107

is the correct?

(w - 2)

7

(w – 5) 3

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1. To show how to multiply out (remove) and simplify an expression that has two single brackets.

1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with two expression with two single brackets.single brackets.

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2.2. Be able multiply out a Be able multiply out a expression with two single expression with two single brackets and the simplify.brackets and the simplify.


Removing Two Single Removing Two Single BracketsBrackets

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4(m - 3) - (m + 2) =4m - 12 - m - 2Example 1

Example 2


= 3m - 14

Tidy Up

7(y - 1) - 2(y + 4) =7y - 7 - 2y - 8 = 5y - 15

Tidy Up


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3 x 14 – 8= 34

A = 4(y – 5)

Example 3 : Find an expression for the difference inthe areas.


(y - 5)

47 A = 7(y- 4)

(y- 4)Difference = 7(y - 4) - 4(y – 5)

= 7y - 28 - 4y + 20= 3y - 8

Calculate difference if

y = 14


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2 x 4 + 20=28

A = 7(x + 1)

Example 3 : Find an expression for the difference inthe areas.


(x + 1)

79 A = 9(x + 3)

x + 3Difference = 9(x + 3) - 7(x + 1) = 9x + 27 – 7x - 7

= 2x + 20Calculate

difference if x = 4



Now try Ex 3.2

Ch3 MIA (page 46)

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Apr 22, 2023Apr 22, 2023


w.m

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(a) x + 3 = 8 (b) 2x – 14 = 50

Q2. Is this statement true

(x – 1) – 3(x + 1) = -2x

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Q3.1 13 12 3

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1. To show how to solve equations that have bracket terms.

1.1. Be able to multiply out Be able to multiply out brackets and solve brackets and solve equations.equations.

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Equations and Equations and bracketsbrackets

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5(x - 3) =25 5x

Example 1


- 15= 25

Multiply out the bracket first and then solve.


5x = 25 + 15 x = 40 ÷ 5

= 40= 8

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6(x - 2) = 3(x + 1) 6x

Example 2


- 12 = 3x


= 3 + 3

6x – 3x -123x -12 = 3

3x = 3 + 12 3x = 15 x = 15 ÷ (3)

= 5

Solve as normalSubstitute

value into original

equation to check answer

18

18

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-3(2 + 2y) =6 – (y + 2) -6

Example 3


- 6y = 6


= 4 - y - y - 2

-6 – 6y-6 - 6y + y = 4

-6 – 5y = 4 – 5y =

10 y = 10 ÷ (-5)

= -2

Tidy up

RHS

Solve as normal

Substitute value into original


6 6


Now try Ex 4.1 & Ex 4.2

Ch3 MIA (page 47)

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Apr 22, 2023Apr 22, 2023


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Q2. Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1

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Q3.1 372 4

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1. To show how to multiply out a pair of brackets using a multiplication table.

1.1. Understand the keypoints Understand the keypoints of multiplying out paired of multiplying out paired brackets.brackets.

Int 2

2.2. Be able to multiply out Be able to multiply out paired brackets using a paired brackets using a multiplication table.multiplication table.


Removing Removing Double BracketsDouble Brackets

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Below is a floor plan of the down stairs of a house.


RemovingRemovingDouble BracketsDouble Brackets

y 5

y

3Tidy up !

y2+ 8y + 15

Living Room

KitchenBathroom

Bedroom

What is the total area of the

downstairs house?

y2 5y

3y 15

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Multiplication table for brackets



(y + 3)(y + 5) y+ 3 y+ 5

Double brackets are used in AREA calculations

+5y+15+3y

y2 Tidy up !

y2+ 8y + 15

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Example 2



(2x - 1)(x + 3) 2x - 1 x+ 3

Be careful with the negative

signs

+6x

-3-x

2x2 Tidy up !

2x2+ 5x - 3

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Example 3



(x + 4)(x2 + 3x + 2) x+ 4 x2 + 3x

Just a biggerMultiplicatio

n Table

+3x2

+12x+4x2

x3 Tidy up !

x3

+ 2

+2x

+8 + 7x2+ 14x+ 8


Now try Ex 5.1First column in each

Question

Ch3 MIA (page 49)

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Removing Removing Double BracketDouble Bracket

Apr 22, 2023Apr 22, 2023


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statement True or False?

Q2. Why does (w + 2)(w + 1) = w2 + 3w + 2

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Q3.1 254 7

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1. To show how to multiply out a pair of brackets using FOIL.

1.1. Understand the keypoints Understand the keypoints of multiplying out double of multiplying out double brackets using FOIL.brackets using FOIL.

Int 2

2.2. Be able to multiply out Be able to multiply out double brackets using double brackets using FOIL.FOIL.



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We can use the F O I L method to multiply out DOUBLE brackets.



Simply remember the word

F O I LMultiply First 2

Multiply Last 2

Multiply Outside 2

Multiply Inside 2

FOIL Method

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(x + 1)(x + 2)

x2 + 2x

Example 1 : Multiply out the brackets and Simplify


1. Write down F O I L+ x + 2

2. Tidy up ! x2 + 3x + 2


Try again using multiplication

table

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(x - 1)(x + 2)

x2 + 2x

Example 2 : Multiply out the brackets and Simplify



1. Write down F O I L- x - 2

2. Tidy up ! x2 + x - 2

Try again using multiplication

table

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FOIL method is very limited

therefore from now on we will use

the multiplication table method.


Squaring Squaring BracketsBrackets

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(x + 1)(x - 2)


(x - 1)(x - 2) (x + 3)(x + 2) (x - 3)(x + 2) (x + 3)(x - 2)

x2 - x - 2 x2 - 3x + 2 x2 + 5x + 6 x2 - x - 6 x2 + x - 6


Now try Ex 5.2First column in each

Question

Ch3 MIA (page 50)

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Apr 22, 2023Apr 22, 2023


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Q2. Multiply out using a multiplication table

(w + 2)(w2 - 2w + 1)

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Q3. Which sums are true1 1 12 2 4 2

1 1 1y y y

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1. To show how to multiply out squared brackets using Multiplication Table.

1.1. Understand the keypoints Understand the keypoints of multiplying out squared of multiplying out squared brackets using brackets using Multiplication Table..

Int 2

2.2. Be able to multiply out Be able to multiply out squared brackets using squared brackets using Multiplication Table..



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= (x + 1)(x + 1) Example 1 : (x + 1)2Multiply out the brackets and Simplify



(x + 1)(x + 1) x+ 1 x+ 1

+ x+ 1+ x

x2 Tidy up !

x2+ 2x + 1

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Example 2 : (x + y)2Multiply out the brackets and Simplify



= (x + y)(x + y) (x + y)(x + y) x+ y x+ y

+xy+ y2+xy

x2 Tidy up !

x2+ 2xy +y2

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(x - 1)2 (x - 2)2

(x - 4)2 (2x – 1)

x2 - 2x + 1 x2 - 4x + 4 x2 - 8x + 16 4x2 - 4x + 1



Now try Ex 6.1 & 6.2

First column in each Question

Ch3 MIA (page 50)

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Apr 22, 2023Apr 22, 2023


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Q2. The sun is 92 million mile away from the earth. Write this in standard form.

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Q3. Is it true1 1 52 3 6

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1. To show how to solve equations that have paired or squared brackets.

1.1. Be able to solve equations Be able to solve equations that contain paired or that contain paired or squared brackets.squared brackets.

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(x + 2)2 =x2 + 82

(x + 2)(x + 2)Example 1


= x2 + 64



x2 + 4x + 4

4x = 60x = 60 ÷ 4 = 15

X table= x2 + 64

4x + 4= 64



289 289

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m(m + 2) =m2 + 10 m2 + 2m

Example 2


= m2 + 10



2mm = 10 ÷ 2 = 5

= 10Substitute value into original


35

35

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6(x - 2) = 3(x + 1) 6x

Example 3


- 12 = 3x


= 3 + 3

6x – 3x -123x -12 = 3

3x = 3 + 12 3x = 15 x = 15 ÷ (3)

= 5

Solve as normalSubstitute

value into original


18

18

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-3(2 + 2y) =6 – (y + 2) -6

Example 4


- 6y = 6


= 4 - y - y - 2

-6 – 6y-6 - 6y + y = 4

-6 – 5y = 4 – 5y =

10 y = 10 ÷ (-5)

= -2

Tidy up

RHS

Solve as normal



6 6

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Solve in the usual

way

A = (x + 7)(x – 2)

Example 5 : The two areas are equal. Find the value of x.


(x + 7)(x – 2)(x + 2)A = (x + 2)2

(x + 2) (x + 2)2 = (x + 7) (x - 2) x2 + 4x + 4

x = 18

X table


= x2 + 5x - 14 4x + 4= 5x - 14

4= x - 14



400 400


Now try Ex 7.1 & 7.2

Ch3 MIA (page 55)

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Documents

Algebraic Operations