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To recall how to expand pairs of brackets for a quadratic
To be able to factorise quadratics in the form 2 + b + c
To be able to solve quadratics in the form 2 + b + c = 0.
OBJECTIVES
Today, we are dealing with a certain form of polynomial. Each has a special name.
DEFINITIONS
4 This is a βconstantβ. It doesnβt change. Itβs also a monomial (one term).
4 + 3 This is βlinearβ.
42 + 3 - 2 This is a βquadraticβ.
23 - 42 + 3 - 2 This is a βcubicβ.4 + 23 - 42 + 3 - 2 This is a βquarticβ.
75 + 4 +23 - 42 + 3 - 2 This is a βquinticβ.
To expand a pair of brackets representing a quadratic, we multiply each term inside each bracket by each term in the other bracket.Here are the combinations.
EXPANDING BRACKETS
(π+π)(π+π)ΒΏππ+ΒΏππ+ΒΏππ+ΒΏππ
Notice that ab and cd are not combinations.
Try the example below.
EXPANDING BRACKETS
ExampleExpand (2 β 1)(4 + 6).
(2 - 1)(4 + 6)=82 +12 -4 -6=82 + 8 - 6
Try the worksheet!
ANSWERS
π₯2+4 π₯+3π₯2β2 π₯β8π₯2β10 π₯+21
π₯2+10 π₯+254 π₯2+10 π₯+4
15 π₯2β19 π₯+64 π₯2+9π₯β924 π₯ 2+46 π₯β1816 π₯2+64 π₯+64β2 π₯2+7 π₯β3ΒΌ π₯2β4 π₯+16
Placing a quadratic into a pair of brackets is called βfactorisationβ. This is the opposite of expanding brackets and more difficult to do.
FACTORISATION
Letβs try a linear expression.
ExampleFactorise 9 β 6.
9 - 6
What is the largest factor that divides into 9 and 6?
3=3( )3 - 2
The contents of the bracket is divided by the coeffi cient outside.
FACTORISATION
A quadratic is more diffi cult.
ExampleFactorise + 5x + 6.
+ 5 + 6
We need to think of two numbers which add together to make 5 and multiply to make 6.
2 and 3
=( ) ( )π₯ π₯
The term has a coeffi cient of 1 because has a coeffi cient of 1.
Each bracket contains .2 and 3 are positive so we get + 2 and + 3.
+ 2 + 3If you are unsure itβs right, expand it out to check!
FACTORISATION
Letβs try another example.
ExampleFactorise - 5x - 36.
We need to think of two numbers which add together to make -5 and multiply to make -36.
Hint: 9 β 4 is 5 and 9 x 4 is 36.
-9 and 4
-9 + 4 = -5-9 x 4 = -36
=( )( )π₯ π₯- 9 + 4We will only look at quadratic expressions where the coeffi cient of is 1. Try the worksheet!
ANSWERS
(π₯+3)(π₯+1)(π₯+3)(π₯+5)(π₯+3)(π₯+4 )(π₯+2 )2(π₯+7 )2
(π₯β7 )(π₯β1)(π₯β5 )2(π₯+7)(π₯+9)(π₯+1)(π₯+8)(π₯β8)(π₯+2)(π₯+20)(π₯+4 )
SOLVING QUADRATIC EQUATIONS THROUGH FACTORISATION
We now know how to factorise quadratics. But how do we solve them for f() = 0?[f() means a function of .]
ExampleSolve - 6x + 5 = 0.
( )( ) = 0π₯ π₯- 5 - 1
This means that β 5 = 0 and β 1 = 0.So = 5 or = 1.
SOLVING QUADRATIC EQUATIONS THROUGH FACTORISATION
Example
Solve + 18x + 72 = 0.( )( ) = 0π₯ π₯+ 6 +
12So .
Try the last worksheet!