3: quadratic expressions expanding brackets and factorisation © christine crisp “teach a level...
DESCRIPTION
Quadratic Expressions e.g. 1 e.g. 2 Expanding Quadratic Expressions A square of a quantity means multiply by itselfTRANSCRIPT
3: Quadratic 3: Quadratic Expressions Expanding Expressions Expanding
Brackets andBrackets andFactorisationFactorisation
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”Vol. 1: AS Core Vol. 1: AS Core
ModulesModules
Quadratic Expressions
Module C1
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Quadratic Expressions
))(( 33 xx
e.g. 1 ))(( 25 xx
10
1032 xxx2 x52x
9
962 xxx3 x32x
e.g. 2
23)( x
Expanding Quadratic Expressions
A square of a quantity means multiply by itself
Quadratic Expressions
12342 xxx
252 2 xx
122 xx202 xx
)4)(4( xx
1682 xx
)2)(12( xx
1.
3.
2.
4.
)4)(3( xx
)5)(4( xx
2)4( x
Exercises
Quadratic Expressions
)( 3 xx
xxx 3e.g. xx 32 mean
s
so, x is a factor of both terms. It is a common factor
xx 32 So,
Factorising Method 1: Common Factors
Common factor
Quadratic Expressions
1. xx 52
xx 63 2
842 2 xx
)( 23 xx
)( 5 xx
)( 422 2 xx
Factorise the following by taking out the common factors
2.
3.
Exercises
Quadratic Expressions
)( 32 xx
)( 4 xx
)( yx 34
4. xx 42
23 3xx
yx 124
963 2 xx
)()( baybax 25
)( 323 2 xx
))(( yxba 25
5.
6.
7. 8.
Quadratic Expressions
))(( 33 xxe.g.1
92 x
One square
A minus sign
e.g.2
22 254 yx ))(( yxyx 5252
(Square roots)
Another square
FactorisingMethod 2: The difference of two squares
Quadratic Expressions
)( 2413 x
))(( yxyx 4343
))(( yy 99
))(( 3232 xx
))(( xx 21213
What about ? 42 x
1.
94 2 x
22 169 yx
281 y
2. 3.
4.
2123 x
Exercises
Can’t do it! It’s NOT a difference
Think!
Common factor first!
Quadratic Expressions
) )( 14 xx(14
x4x1
We need –3x so we want – 4x and +1x.
22 or
432 xxe.g. 1
Factorising
–3x is called the linear term
Method 3: Trinomials
The factors could not give 3 for the coefficient of x, so we try
22 14
Quadratic Expressions
672 xxe.g. 216 32
))(( 16 xx
Constant positive Signs of factors are the same
Method 3: Trinomials
Quadratic Expressions
Constant positive Signs of factors are the same
652 xx
652 xx
1.
2.
3. 1272 xx
))(( xx
))(( xx
))(( xx
2 3
32
3 4
Exercises
Quadratic Expressions
Constant negative Signs of factors are different
652 xx
652 xx
4.
5.
6. 322 xx
))(( xx
))(( xx
))(( xx
1 6
6 1
31
Exercises
Quadratic Expressions
652 xx
2142 xx
7.
1242 xx
1282 xx
15164 2 xx
15143 2 xx
))(( 32 xx
))(( 62 xx
))(( 37 xx
))(( 3252 xx
))(( 353 xx
))(( 26 xx
8.
9.
11.
12.
10.
Exercises
Quadratic Expressions SUMMAR
YThere are 3 methods of factorising quadratic expressions. Common
factors. The difference of 2 squares. Trinomial
factors.• Constant term positive signs are the
same.
• Constant term negative one sign is positive and one is negative.
• Choose a pair of factors of the constant and check the linear term is correct. If not, try again.
• List possible pairs of factors of the constant.
Quadratic Expressions
Quadratic Expressions
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Quadratic Expressions SUMMARY
There are 3 methods of factorising quadratics. Common
factors. The difference of 2 squares. Trinomial
factors.• Constant term positive signs are the
same.
• Constant term negative one sign is positive and one is negative.
• Choose a pair of factors of the constant and check the linear term is correct. If not, try again.
• List possible pairs of factors of the constant.
Quadratic Expressions Examples
1. xx 52 xx 63 2
842 2 xx2. 3. 4. xx 42
23 3xx yx 124
963 2 xx
)()( baybax 25
5. 6. 7. 8. 9. 94 2 x
281 y10.
)( 23 xx)( 5 xx
)( 422 2 xx
)( 32 xx)( 4 xx
)( yx 34
)( 323 2 xx))(( yxba 25
))(( yy 99))(( 3232 xx
Quadratic Expressions Examples
22 169 yx 11. 12.
2123 x
652 xx652 xx
13.14.
1272 xx2142 xx
1242 xx
1282 xx
15164 2 xx15143 2 xx
15.16.17.18.19.20.
))(( yxyx 4343 ))(( xx 21213 )( 2413 x
))(( 32 xx
))(( 62 xx
))(( 37 xx
))(( 3252 xx))(( 353 xx
))(( 26 xx
)3)(2( xx
)3)(4( xx