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Algebraic manipulations
SLIDE 1
The value of is how much more than the value of ?
A 8
B 14
C
D
E
SLIDE 2
The value of is how much more than the value of ?
Solution
B 14
SLIDE 3
If what is the value of ?
A
B 3
C
D 19
E 192
SLIDE 4
If what is the value of ?
Solution
B 3
SLIDE 5
Which of the following is equivalent to ?
A
B
C
D
E
SLIDE 6
Solution
E
SLIDE 7
Which of the following expressions is equivalent to the one above?
A
B
C
D
SLIDE 8
Solution
A
SLIDE 9
Example
By expanding the left-hand side and equating coefficients, find a, b and c.
Solution
SLIDE 10
By equating coefficients, find a, b and c.
SLIDE 11
SLIDE 12
The sum of and can be written in the form , where a, b and c are constants. What is the value of ?
SLIDE 13
The sum of and can be written in the form , where a, b and c are constants. What is the value of ?
Solution
SLIDE 14
Given , find .
SLIDE 15
SLIDE 16
? Factorise
? Let and . By substituting into your answer to part ?, find .
SLIDE 17
?
? Given and and , then
Note,
SLIDE 18
The expression can be written in the form , where a and b are constants. Find .
SLIDE 19
The expression can be written in the form , where a and b are constants. Find .
Solution
Hence
SLIDE 20
Factorise
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? Find without a calculator
? Find without a calculator
SLIDE 21
Factorise
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SLIDE 22
Factorise
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SLIDE 23
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SLIDE 24
Factorise
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SLIDE 25
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Note. In the last by comparison with , which is the difference of two squares, where and .
SLIDE 26
Singularity
No function is defined where a denominator is equal to zero. For example, the function is undefined when , which is when .
The reason for this is that in algebra it is never permitted to divide by zero. Any value where a function is undefined is called a singularity.
SLIDE 27
State the values for which the following functions are undefined
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SLIDE 28
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SLIDE 29
What is invalid (incorrect, wrong) about the following?
What should be the correct solution to the equation?
SLIDE 30
Strictly, we should write provided that .
where is short for �x is not equal to 1�.
Incorrect Correct
We cannot have a solution because is undefined at . The correct and only answer is .
SLIDE 31
Which of the following are all the solutions to the above?
A
B
C
D
SLIDE 32
Solution
C only
SLIDE 33
? Substitute and into , taking only the positive square root in each case.
Consider the following
? Substitute into each line of the above, taking and state whether the line is true or false.
? Substitute into each line of the above, allowing and state whether the line is true or false.
? State the solutions of .
SLIDE 34
? Substituting and into and taking only the positive square root
? Substituting into each line
? Substituting into each line
? In fact, both and are solutions to because can be negative.
SLIDE 35
Someone argues as follows
On substituting into we obtain
Hence, cannot be a solution to .
Explain what is wrong with this argument and show that is a solution to
SLIDE 36
The error in the argument is that . So, we have
To show that is a solution to
SLIDE 37
Which of the following is the solution set to ?
A
B
C
D
E
SLIDE 38
However, is not a solution to the original equation.
Substitution of gives and . The reason why is not a solution is that in we cannot have a negative value under the root symbol. Thus . Hence the negative is not a solution. The answer is B, only.
SLIDE 39
Simplify
? ?
? ?
SLIDE 40
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SLIDE 41
Factorize
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SLIDE 42
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SLIDE 43
Find and simplify
SLIDE 44
SLIDE 45
Find a, b and c for each of the following
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SLIDE 46
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SLIDE 47
Find a, b and c for each of the following
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SLIDE 48
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SLIDE 49
Example
By substituting into solve
Solution
SLIDE 50
By substituting or otherwise solve
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SLIDE 51
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SLIDE 52
If n is an integer, prove that the sum of and is a square number.
SLIDE 53
If n is an integer, prove that the sum of and is a square number.
Solution
SLIDE 54
Let n be an integer. Prove that
is the product of three consecutive number.
SLIDE 55
SLIDE 56
Example
Prove algebraically that
Solution
Let , then
In general
If is an infinitely recurring decimal expansion,
then .
If is an infinitely recurring decimal expansion,
then .
If is an infinitely recurring decimal expansion,
then .
Likewise, for any recurring decimal expansion.
SLIDE 57
? Find the fraction equivalent to , expressing this fraction in its lowest terms.
? Prove algebraically that is the fraction you found.
SLIDE 58
?
? To prove algebraically that
Proof
Let , then
SLIDE 59
Prove algebraically that
SLIDE 60
Prove algebraically that
Solution
SLIDE 61
A sphere has been cut into four equal quarters. The volume of each quarter is . Find the surface area of each quarter, giving your answer to 3 significant figures.
SLIDE 62
A sphere has been cut into four equal quarters. The volume of each quarter is . Find the surface area of each quarter, giving your answer to 3 significant figures.
Solution
The surface is in three parts. The �outer� part is of the surface of the sphere, and the two �inner� parts are both semicircles passing through the centre of the sphere.
SLIDE 61B
A sphere has been cut into four equal quarters. The volume of each quarter is . Find the surface area of each quarter, giving your answer to 3 significant figures.
SLIDE 62B
A sphere has been cut into four equal quarters. The volume of each quarter is . Find the surface area of each quarter, giving your answer to 3 significant figures.
Solution
The surface is in three parts. The �outer� part is of the surface of the sphere, and the two �inner� parts are both semicircles passing through the centre of the sphere.
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