Algebraic Operations

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Algebraic Operations
Summary of Factorising Methods
Introduction to Quadratic Equation
Factorising Trinomials (Quadratics)
Real-life Problems on Quadratics
Finding roots by factorising and formula
Exam Type Questions
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Starter Questions
Nat 5
Q1. Remove the brackets (a) y(4y 3x) (b) (x
5)(x - 5)
Q2. For the line y -x 5, find the
gradient and where it cuts the y axis.
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Q3. Find the highest common factor for p2q and
pq2.
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Factorising
Methods
Nat 5
Learning Intention
Success Criteria

1. To be able to identify the three methods of
   factorising.

1. To review the three basic methods for factorising.

1. Apply knowledge to problems.

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Summary of Factorising
Nat 5
When we are asked to factorise there is priority
we must do it in.

• Take any common factors out and put them
• outside the brackets.

2. Check for the difference of two squares.
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3. Factorise any quadratic expression left.
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Common Factor
Nat 5
Factorise the following
2x(y 1)
(a) 4xy 2x (b) y2 - y
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y(y 1)
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Difference of Two Squares
Nat 5
When we have the special case that an expression
is made up of the difference of two squares
then it is simple to factorise
The format for the difference of two squares
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a2 b2
First square term
Second square term
Difference
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Difference of Two Squares
Check by multiplying out the bracket to get back
to where you started
Nat 5
a2 b2
First square term
Second square term
Difference
This factorises to
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( a b )( a b )
Two brackets the same except for and a -
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Difference of Two Squares
Nat 5
Keypoints
Format a2 b2
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Always the difference sign -
( a b )( a b )
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Difference of Two Squares
Nat 5
Factorise using the difference of two squares
( w z )( w z )
(a) w2 z2 (b) 9a2 b2 (c) 16y2 100k2
( 3a b )( 3a b )
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( 4y 10k )( 4y 10k )
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Difference of Two Squares
Nat 5
Factorise these trickier expressions.
6(x 2 )( x 2 )

• (a) 6x2 24
• 3w2 3
• 8 2b2
• (d) 27w2 12

3( w 1 )( w 1 )
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2( 2 b )( 2 b )
3(3 w 2 )( 3w 2 )
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Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (2) and Diagonals sum to give middle
value 3x.
x2 3x 2
x
2
2
x
(2) x( 1) 2
1
x
1
x
(2x) ( 1x) 3x
( ) ( )
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Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (5) and Diagonals sum to give middle
value 6x.
x2 6x 5
x
5
5
x
(5) x( 1) 5
1
x
1
x
(5x) ( 1x) 6x
( ) ( )
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Both numbers must be -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (4) and Diagonals sum to give middle
value -4x.
x2 - 4x 4
x
- 2
- 2
x
(-2) x( -2) 4
- 2
- 2
x
x
(-2x) ( -2x) -4x
( ) ( )
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One number must be and one -
Factorising Using St. Andrews Cross method
Strategy for factorising quadratics
Find two numbers that multiply to give last
number (-3) and Diagonals sum to give middle
value -2x
x2 - 2x - 3
x
- 3
- 3
x
(-3) x( 1) -3
1
x
1
x
(-3x) ( x) -2x
( ) ( )
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Factorising Using St. Andrews Cross method
Nat 5
Factorise using SAC method
(m 1 )( m 1 )

• (a) m2 2m 1
• y2 6m 5
• b2 b -2
• (d) a2 5a 6

( y 5 )( y 1 )
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( b - 2 )( b 1 )
( a - 3 )( a 2 )
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.1 Ch14 (page132)
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Starter Questions
Nat 5
Q1. True or false y ( y 6 ) -7y y2 -7y 6
Q2. Fill in the ? 49 4x2 ( ? ?x)(? 2?)
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Q3. Write in scientific notation 0.0341
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This is called a quadratic equation
Quadratic Equations
Nat 5
A quadratic function has the form
a , b and c are constants and a ? 0
f(x) a x2 b x c
The graph of a quadratic function has the basic
shape
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y
y
The x-coordinates where the graph cuts the x
axis are called the Roots of the function.
x
x
i.e. a x2 b x c 0
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Quadratic Equations
h
This is the graph of a golf shot.
The height h m of the ball after t seconds is
given by
h 15t 5t2
The graph of a quadratic function is called a
parabola
(a) For what values t does h 0
t
(b) What are the solutions for
t 0
t 3
15t 5t2 0
t 0
t 3
and
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Quadratic Equations
h
This is the graph of a parabola
h 10t 2t2
(a) From the graph, what are the roots of the
quadratic eqn.
Both 8
h 10t 2t2
(b) What is the value of h for t 1 and t 4
t
t 0
t 5
(c) What are the solutions of the
quadratic equation
10t 2t2 0
t 0
t 5
and
(d) What is the solution of the quadratic
equation
10t 2t2 12.5
2.5
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.2 Ch14 (page133)
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Starter Questions
Nat 5
Q1. Multiple out the brackets and
simplify. (a) ( 2x 5 )( x 5 )
Q2. Find the volume of a cylinder with height
6m and diameter 9cm
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Q3. True or false the gradient of the line is 1 x
y 1
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Factors and Solving Quadratic Equations
Nat 5
Learning Intention
Success Criteria

1. Be able find factors using the three methods to
   solve quadratic equations.

1. To explain how factors help to solve quadratic
   equations.

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Factors and Solving Quadratic Equations
Nat 5
The main reason we learn the process of
factorising is that it helps to solve (find
roots) for quadratic equations.
Reminder of Methods

• Take any common factors out and put them
• outside the brackets.

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2. Check for the difference of two squares.
3. Factorise any quadratic expression left.
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Solving Quadratic Equations
Examples
Nat 5
Solve ( find the roots ) for the following
Common Factor
16t 6t2 0
Common Factor
x2 4x 0
2t(8 3t) 0
x(x 4) 0
x - 4 0
2t 0
and
8 3t 0
x 0
and
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x 4
t 8/3
t 0
and
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Solving Quadratic Equations
Examples
Nat 5
Solve ( find the roots ) for the following
100s2 25 0
x2 9 0
Difference 2 squares
Difference 2 squares
(10s 5)(10s 5) 0
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(x 3)(x 3) 0
10s 5 0
and
10s 5 0
x -3
x 3
and
s - 0.5
s 0.5
and
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.3 Ch14 (page134)
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Solving Quadratic Equations
Examples
Nat 5
Common Factor
2x2 8 0
80 125e2 0
Common Factor
2(x2 4) 0
5(16 25e2) 0
Difference 2 squares
Difference 2 squares
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5(4 5e)(4 5e) 0
2(x 2)(x 2) 0
(4 5e)(4 5e) 0
(x 2)(x 2) 0
4 5e 0
and
4 5t 0
(x 2) 0
and
(x 2) 0
x 2
and
x - 2
e - 4/5
e 4/5
and
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.4 upto Q12 Ch14 (page135)
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Solving Quadratic Equations
Examples
Nat 5
Solve ( find the roots ) for the following
x2 5x 4 0
1 x - 6x2 0
SAC Method
SAC Method
x
1
4
3x
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x
1
1
-2x
(x 4)(x 1) 0
(1 3x)(1 2x) 0
x 4 0
x 1 0
and
1 3x 0
and
1 - 2x 0
x - 4
and
x - 1
x - 1/3
and
x 0.5
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.4 Q13.... Ch14 (page137)
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Solving Quadratic Equations
Examples
Nat 5
Multiply out and rearrange
Multiply out and rearrange
Solve ( find the roots ) for the following
(x 4)2 36
5x(2x 1) - 10 x(7x 6)
3x2 - x - 10 0
x2 8x - 20 0
SAC Method
SAC Method
3x
5
x
10
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x
- 2
x
-2
(x 10)(x - 2) 0
(3x 5)(x 2) 0
x 10 0
x - 2 0
and
3x 5 0
and
x - 2 0
x - 10
and
x 2
x - 5/3
and
x 2
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Multiply through by 2(x - 1)(x 2) to remove
denominators
Solving Quadratic Equations
Examples
Nat 5
Solve ( find the roots ) for the following
x
- 4
x
1
2(x 2) 2(x 1) (x 1)(x 2)
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2x 4 2x 2 x2 x - 2
(x - 4)(x 1) 0
x2 - 3x 4 0
x - 4 0
x 1 0
and
SAC Method
x 4
and
x - 1
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Multiply through by x(x 1) to remove
denominators
Solving Quadratic Equations
Examples
Nat 5
Solve ( find the roots ) for the following
x
3
x
- 2
6(x 1) - 6x x(x 1)
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6x 6 6x x2 x
(x 3)(x - 2) 0
x2 x 6 0
x 3 0
x - 2 0
and
SAC Method
x - 3
and
x 2
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.5 Ch14 (page138)
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Starter Questions
Nat 5
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Real-life Quadratics
Nat 5
Learning Intention
Success Criteria

1. To be able to using quadratic theory in real-life
   problem.

1. To show how quadratic theory is used in real-life.

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Real-life Problems
A rectangle garden is twice as long as it is
wide. The area is 200m2. Find the dimensions of
the rectangle garden.
Let width be x
Area length x breadth
Length is 2x
200 2x x x
200 2x2
x2 100
x 10
x -10
and
x must be positive ( We cannot get a negative
length !!! )
Width is equal to 10m
Length is equal to 20m
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Exam Type Questions
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Real-life Problems
The height in metres of a rocket fired vertically
upwards is give by the formula
h 176t 16t2
(a) When will the rocket be at a height of 160
metres.
160 176t 16t2
16t2 - 176t 160 0
t2 - 11t 10 0
(t 10)(t 1) 0
t 10
and
t 1
(b) Is it possible for the rocket to h 500
metres.
Since 500 176t -16t2 has no solution not
possible.
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Factorising Methods
Nat 5
Now try N5 TJ Ex 14.6 Ch14 (page139)
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Starter Questions
Nat 5
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Roots Formula
Nat 5
Learning Intention
Success Criteria

1. To be able to solve quadratic equations using
   quadratic formula.

1. To explain how to find the roots (solve)
   quadratic equations by use quadratic formula.

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Roots Formula
Nat 5
Every quadratic equation can be rearranged into
the standard form
a, b and c are constants
ax2 bx c 0
Examples find the constants a, b and c for the
following
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3x2 x 4 0
a 3
b 1
c 4
x2 - x - 6 0
a 1
b -1
c -6
x(x - 2) 0
x2 2x 0
a 1
b -2
c 0
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Roots Formula
Nat 5
Now try MIA Ex6.1 First Column (page 166)
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Roots Formula
Nat 5
Every quadratic equation can be rearranged into
the standard form
a, b and c are constants
ax2 bx c 0
In this form we can use the quadratic root
formula to find the roots.
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Roots Formula
Nat 5
Example Solve x2 3x 3 0
ax2 bx c 0
1
3
-3
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Roots Formula
Nat 5
and
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and
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Roots Formula
Use the quadratic formula to solve the following
2x2 4x 1 0
x2 3x 2 0
x -1.7, -0.3
x -3.6, 0.6
5x2 - 9x 3 0
3x2 - 3x 5 0
x 1.9, -0.9
x 1.4, 0.4
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Roots Formula
Nat 5
Now try MIA Ex7.1 7.2 (page 168)
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Exam Type Questions
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Exam Type Questions
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Exam Type Questions
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Exam Type Questions
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Exam Type Questions