Quadratic Expressions Worksheets Page 5
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36number and algebra
(a + b)
= a
+ ab + ab + b
2
2
2
= a
+ 2ab + b
2
2
• Similarly (a − b)
= a
− 2ab + b
2
2
2
.
• This expansion is often memorised. To fi nd the square of a binomial:
– square the fi rst term
– multiply the two terms together and then double them
– square the last term.
WOrKed eXamPle 3
WOrKed eXamPle 3
WOrKed eXamPle 3
Expand and simplify each of the following.
−3(2x + 7)
(2x − 5)
2
2
a
b
THInK
WrITe
(2x − 5)
2
a
1
Write the expression.
a
2
Expand using the rule (a − b)
= a
− 2ab + b
= (2x)
− 2 × 2x × 5 + (5)
2
2
2
2
2
.
= 4x
− 20x + 25
2
−3(2x + 7)
2
Write the expression.
b
1
b
= −3[(2x)
+ 2 × 2x × 7 + (7)
2
2
Expand the brackets using the rule
]
2
= −3(4x
+ 28x + 49)
(a + b)
= a
+ 2ab + b
2
2
2
2
.
= −12x
2
− 84x − 147
Multiply every term inside the brackets by the term
3
outside the brackets.
The difference of two squares
• When a + b is multiplied by a − b (or vice‐versa),
(a + b) (a − b) = a
2
− ab + ab − b
2
= a
− b
2
2
The expression is called the difference of two squares and is often referred to as DOTS.
This result can be memorised as a short cut.
WOrKed eXamPle 4
WOrKed eXamPle 4
WOrKed eXamPle 4
Expand and simplify each of the following.
(3x + 1) (3x − 1)
4(2x − 7) (2x + 7)
a
b
THInK
WrITe
(3x + 1) (3x − 1)
Write the expression.
a
1
a
2
Expand using the rule (a + b) (a − b) = a
− b
= (3x)
− (1)
2
2
2
2
.
= 9x
2
− 1
Maths Quest 10 + 10A
272
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