GCSE Maths

Numbers
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Integers
Prime Numbers and Composite Numbers
Negative Numbers
Comparing Numbers Using Symbols
The Square Numbers
The Laws of Indices
Negative Indices
The Cube Numbers
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
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Vectors
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Vectors
Probability
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Introduction to probability
Finding the sum of probabilities
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
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Population and samples
Representing data
Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
Quadratic Equations
Mathematical Formulae
Algebraic Equivalence and Proof
Functions
Algebra
Sequences
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Generating sequences
Nth Term Sequences
Sequences
Ratio, Proportion and Rates of Change
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Inverse Proportion
Conversion
Scale Factors
Percentages
Direct Proportion
Rate, Ratio and Proportion Examples
Ratios
Ratios
Ratio, Proportion and Rates of Change
Geometry and Measures
9 Topics
Loci and constructions
Properties of angles
Properties of polygons
Transformations
Enlargement
Circle theorems
Plane figure properties
Plans and elevations
Areas and Volumes in Map Scale Problems
Mensuration and Calculation
10 Topics
Map scales
Bearings
Area
Volume
Circles: arcs and sectors
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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The Difference of Two Squares

GCSE Maths Algebra The Difference of Two Squares

Note that (a+b)(a+b) = (a+b)^{2} and (a-b)(a-b) = (a-b)^{2}.

(a+b)(a-b) = a^{2} - b^{2} is known as the difference of two squares. Often multiplying two numbers becomes easier when using algebraic expression. Let’s look at how to use them to help solve arithmetic problems.

Example

Find 57 \times 23

Write 57 as 50 + 7 and 23 as 20 + 3.

57 \times 23 = (50 + 7)(20 + 3)

= 50(20) + 50(3) + 7(20) + 7(3)

= 1000 + 150 + 140 + 21

= 1311

Or, we could have said 57 = 60 - 3 and 23 = 20 + 3, to obtain 57 \times 23 = (60 - 3)(20 + 3)

= 60(20) + 60(3) - 3(20) - 3(3)

= 1200 + 180 - 60 - 9

= 1311

Or, 57=60-3 and 23=30-7 to obtain:

57 \times 23 = (60-3)(30-7)

= 60(30) - 60(7) - 3(30) - 3(-7)

= 1800 - 420 - 90 + 21

= 1311

Example

Find { 63 }^{ 2 }

We solve this in two ways. First write 63 as 60 + 3. then use { (a + b) }^{ 2 } = { a }^{ 2 } + 2ab + { b }^{ 2 }. Also, we write 63 as 70 - 7 and use (a - b)^{ 2 } = { a }^{ 2 } - 2ab + { b }^{ 2 }.

We should arrive at the same conclusion:

{ 63 }^{ 2 } = { 60 + 3 }^{ 2 } = { 60 }^{ 2 } + 2(60) (3) + { 3 }^{ 2 }

= 3600 + 360 + 9

= 3969

Or

{ 63 }^{ 2 } = { 70 - 7 }^{ 2 } = { 70 }^{ 2 } - 2(70) (7) + { 7 }^{ 2 }

= 4900 - 980 + 49

= 3969, the same as the other answer above.

Example

Find 37 \times 43

37 \times 43 = (40 - 3) (40 + 3)

= { 40 }^{ 2 } - { 3 }^{ 2 }

= 1600 - 9

= 1591

Example

Find the value of the following

i) 202^{ 2 }

ii) 198^{ 2 }

iii) 202^{2} - 198^{2}

iv) 202 \times 198

i) 202^{ 2 } = (200 + 2)^{2}

= 200^{2} + 2(200)(2) + 2^{2}

= 40000 + 800 + 4

= 40804

ii) { 198 }^{ 2 } = { (200 - 2) }^{ 2 }

= 200^{2} - 2(200)(2) + 2^{2}

= 40000 - 800 + 4

= 39204

iii) 202^{2} - 198^{2} = 40804 - 39204

=1600

or, 202^{2} - 197^{2} = (202+198)(202-198)

=(400)(4)

=1600

iv) 202 \times 198 = (200 + 2) (200 - 2)

= { 200 }^{ 2 } - { 2 }^{ 2 }

= 40000 - 4

= 39996

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