Expanding Brackets
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Words to know
Expand: to get rid of brackets
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Brackets
Invisible x sign
What does 4(a + b) mean?
4 × (a + b)
or
(a + b) + (a + b) + (a + b) + (a + b) OR
=a+b+a+b+a+b+a+b
= 4a + 4b
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OR
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Expanding a bracket
Multiply the number/letter outside the bracket
with everything inside the bracket.
=
X
4(a + b) = 4a + 4b
X
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=
http://www.youtube.com/watch?v=QzlUo_80iQA
(Expand bracket)
OR…
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Grid method to Expand brackets
b
E.g. to expand 4(a
4 a + b)
4b
4a
+
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Only Use 1
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Your Turn: Expand the following
1
2
3
4
5
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Question
Answer
2( x  3)
4 x( x  6)
 2x  6
2( x  3)
 4 x  24 x
2
 2 x  6
2( x  3)
 2 x  6
5 x(2 x  1)
 10 x  5 x
2
Your Turn
Expand these brackets
1.
3(a - 4) = 3a - 12
2.
6(2c + 5) = 12c + 30
-2(d + g) = -2d - 2g
c(d + 4) = cd + 4c
-5(2a - 3) = -10a + 15
a(a - 6) = a2 - 6a
4r(2r + 3) = 8r2 + 12r
- (4a + 2) = -4a - 2
3.
4.
5.
6.
7.
8.
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Expanding larger brackets
Look at this algebraic expression:
–a(2a2 – 2a + 3)
When there is a negative term outside the bracket, the signs of
the multiplied terms change.
–a(2a2 – 3a + 1) = –2a3 + 3a2 – a
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Click on blue rectangles to reveal question
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Questions to do from the books
Achieve
Merit
Excellence
Gamma P12 Ex2.01 Q1 – 16
CAT 1.2
P24 Q157 - 160
Only 4 questions in the CAT 1.2 book.
Previous slide is good for generating questions
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Expand and simplify
For example,
3x + 2(5 – x)
Remember BEDMAS.
First expand the bracket then collect together like terms.
3x + 10 – 2x
= 3x – 2x + 10
(now gather like terms)
(now simplify)
= x + 10
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http://www.youtube.com/watch?v=XQoQRMLyqY&feature=related (expand and simplify)
E.g. Expand and simplify
Simplify
4 –1 (5n – 3)
Multiply inside the bracket by –1 and collect together like terms.
4 – 5n + 3 (now gather like terms)
= 4 + 3 – 5n
= 7 – 5n
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(now simplify)
E.g. Expand and simplify
Simplify
2(3n – 4) + 3(3n + 5)
Multiply out both brackets and collect together like terms.
6n – 8 + 9n + 15
= 6n + 9n – 8 + 15
= 15n + 7
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(now collect like terms)
(now simplify)
E.g. Expand and simplify
Simplify
5(3a + 2b) – 2(2a + 5b)
Multiply out both brackets and collect together like terms.
15a + 10b – 4a –10b
= 15a – 4a + 10b – 10b
= 11a
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(now collect like terms)
(now simplify)
Your Turn: Expand and simplify
5  4(3x  2)
 5  12 x  8
 12 x  3
3(4 x  2)  (5 x  1)  12 x  6  5x  1
Remember
Invisible -1 in front
of the brackets
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 7x  7
Your Turn: Expand and simplify
2(3x  7)  5(2 x  3)  6 x  14  10 x  15
 4 x  29
4 x(2 x  1)  3x(4 x  2)  8 x  4 x  12 x  6 x
2
2
 4 x  10 x
2
4(3x  2)  ( x  5)  4(3x  2)  1( x  5)
Invisible -1 in front of
the brackets!
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 12 x  8  x  5
11x + 13
Click on pairs of equivalent expressions
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Questions to do from the books
Achieve
Gamma
CAT 1.2
Merit
P12 Ex2.01 Q17 – 56
P24 Q168, 172, 173,
174, 175,
Only 5 questions in the CAT 1.2 book.
More questions on next slide
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Excellence
Expand these brackets and simplify wherever possible:
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1.
- (4a + 2) = -4a - 2
2.
8 - 2(t + 5) = -2t - 2
3.
2(2a + 4) + 4(3a + 6) =
16a + 32
4.
2p(3p + 2) - 5(2p - 1) =
6p2 - 6p + 5
Expanding Double Brackets
http://www.youtube.com/watch?v=iB3L5csbB4E
(expand double brackets)
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Expanding double brackets
Invisible x sign
between
brackets
E.g. Expand (3 + t)(4 – 2t)
Multiply every term in the second bracket by every term in the
first bracket.
(3 + t)(4 – 2t) =
12 – 6t + 4t – 2t2
= 12 – 2t – 2t2
This is a quadratic expression as the
highest power of x is 2.
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Using the grid method to expand brackets
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Find the area of the rectangle
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Your Turn: Expand and simplify
1.
(c + 2)(c + 6) = c2 + 8c + 12
2.
(2a + 1)(3a – 4) = 6a2 – 5a – 4
3.
(3a – 4)(5a + 7) = 15a2 + a – 28
4.
(p + 2)(7p – 3) = 7p2 + 11p – 6
5.
(2 x  3)(5 x  4)  10 x 2  8 x  15 x  12
 10 x 2  7 x  12
4) Expand
6.
( x  3)( x  7)
(4 x2 3) 2  (4 x  3)(4 x  3)
 x  7 x 2 3 x  21
 16 x  12 x  12 x  9
 x  4 x 2 21
 16 x  24 x  9
(3 x  1)(5 x  7)
2
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Questions to do from the books
double brackets
Achieve
P14 Ex2.02
Gamma P15 Ex2.03 Q1 – 8,
15 - 36
P24 Q161 – 163,
CAT 1.2
165, 169 – 171, 176
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Merit
P15 Ex 2.03 Q37
P25 Q177 – 180
Excellence
Matching quadratic expressions 1
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Matching quadratic expressions 2
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Squaring Brackets
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Squaring brackets
Expand and simplify: (2 – 3a)2
Rewrite this as,
(2 – 3a)2 = (2 – 3a)(2 – 3a)
Expand
(2 – 3a)(2 – 3a) = 4 – 6a – 6a + 9a2 (simplify)
= 4 – 12a + 9a2
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OR….
Grid method for squaring brackets
Expand and simplify: (2 – 3a)2
(2 – 3a)(2 – 3a)
2
2
–3a
–3a
4
–6a
–6a
9a2
Now simplify:
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4 – 12a + 9a2
Your Turn
1.
 10 x 2  7 x  12
(4 x  3) 2  (4 x  3)(4 x  3)
 16 x 2  12 x  12 x  9
 16 x 2  24 x  9
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2.
(c + 7)2 = c2 + 14c + 49
3.
(4g – 1)2 = 16g2 – 8g + 1
4.
(3x + 5)2 = 9x2 + 30x + 25
5.
(2h – 4)2 = 4h2 – 16h + 16
Squaring expressions
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Questions to do from the books
Achieve
Merit
Gamma P15 Ex2.04
P16 Ex 2.05
Q32 – 44
CAT 1.2
See below and
next slide
P24 Q164, 166, 167
Excellence
Merit: DOTS
Try and find two brackets to multiply together that leave
only two terms when expanded.
Answer ( x  2)( x  2)  x 2  2 x  2 x  4  x 2  4
This is the only situation where we end up with 2 terms
Can you explain why this is called the Difference Of Two Squares?
Give another example of DOTS and show that this works.
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Merit: Expansion of cubics
E.g. 1) Expand the following
Park the first bracket
Simplify
Multiply using lines
Simplify
E.g. 2) Expand the following
( x  2)( x  3)( x  4)
 ( x  2) (x 2 4x 3x 12)
 ( x  2)( x 2  x  12)
 x3  x 2 12x 2x 2 2x 24
 x3  3x 2  10 x  24
( x)( x  3)( x  4)
 ( x)(x 2 4x 3x 12)
 ( x)( x 2  x  12)
 x3  x 2  12 x
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Your Turn: Revision
3) Expand
1. ( x  2)( x  5)  x 2  5 x  2 x  10
 x 2  3 x  10
2. ( x  1)( x  1)  x 2  x  x  1
 x2  1
2
(
x

7)
 ( x  7)( x  7)
3.
5. ( x  4)( x  7)  x 2  7 x  4 x  28
 x 2  3 x  28
6. ( x  3)( x  3)  x  3 x  3 x  9
2
 x2  9
7. ( x  2) 2  ( x  2)( x  2)
 x 2  7 x  7 x  49
 x2  2 x  2 x  4
 x 2  14 x  49
 x2  4 x  4
4. 3  2( 4 x  7)  3  8 x  14
 17  8 x
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3) Expand
8. 4  5(2 x  3)  4  10 x  15
 19  10 x